Math 131AH Bonus point system

* - no more than 20 bonus points per student will be applied to the final grade.

Note: some of the page numbers in later corrections may be off by 1 from the hardcopy version, because the previous corrections may have altered the page length of the notes. The on-line PDF files of the notes are always the most up-to-date, incorporating all the corrections listed below.

• (Mar 10) Nameless pointed out that in the hint for HW9 Q9(b), Lemma 10 should instead read Lemma 11.
• (Mar 7) Anonymous_1 pointed out that HW8 Q9(a) follows easily from an earlier HW question (specifically, Proposition 13 from Week 6). It will be OK to just quote that earlier proposition to do this part; however with all the machinery of Week 9 there is now a better proof (if you may recall, the proof of Proposition 13 of Week 6 was fairly nasty)).
• (Mar 7) Tal and Stefan simultaneously pointed out that on Q8(a) (Proposition 21) of Assignment 9, that the condition a < = b should instead read a < b. Tal also pointed out that for Q10 (Proposition 24), the hypotheses "f and g are differentiable at 0" should read "f and g are differentiable at x_0".
• (Mar 3) Nameless pointed out that in Q9(a) of Assignment 8, n should be specified as a natural number.
• (Feb 26) I discovered that in Q4 of Assignment 8, (proof of the chain rule), it should be stated in the hypotheses of Theorem 16 (the chain rule) that x_0 is an element of X, and y_0 is an element of Y. (Thanks to Nameless for clarifying this correction).
• (Feb 20) Marion pointed out that in the hint to Q8 of Assignment 6, "Proposition 30" should instead read "Lemma 30".
• (Feb 17) Anonymous_1 pointed out that in Q5(b) of Assignment 6 (I.e. Proposition 6(b) on page 11 of Week 6 notes), f(x_0) should read L.
• (Feb 13) Jared noted that in Q5 of Assignment 4, the subscripting in the hint is incorrect (sup_{n=1}^infinity a_n should instead read sup (a_n)_{n=1}^infinity).
• (Feb 12) Tal noted that in Q2(e) of Assignment 5 (i.e. Proposition 4(e) from Week 5 notes), "x^q < x^r" should read "x^q > x^r".
• (Feb 12) Anonymous_1 noted that in Q3(b) of Assignment 5, one needs to assume that M is positive.
• (Feb 10) Nameless noted that in part (b) of Q5 of Assignment 5 (i.e. Proposition 10(b) from Week 5 notes), the right-hand side should be f(x_0), not 0.
• (Feb 2) Evan noted that Q9(a) of Assignment 4 was redundant (it is part of Lemma 28, which is proved in Q8(a)). Hence Q9(a) should be dropped from the assignment (so Q9 now only refers to Q9(b)).
• (Jan 22) Tal and Marion simultaneously pointed out that with the relabeling of Proposition 2 of Week 2 notes, Q7 of Assignment 2 should now refer to Proposition 2(h) rather than Proposition 2(g).
• (Jan 21) Updowncharm pointed out a typo in the hint for Q8 of Assignment 2: "this sequence cannot be epsilon-close" should read "this sequence cannot be epsilon-close to the zero sequence 0,0,0,...".
• (Jan 20) Nameless pointed out that in the hint to Q5 of Assignment 2, both occurrences of "1-close" should have read "1-steady".

• (Mar 18) Anonymous_1 observed that it is possible to prove Lemma 2 using the second fundamental theorem of calculus without requiring Riemann integrability (by applying the second FToC to H = F-G).
• (Mar 17) Nameless observed that alpha(I) should read alpha[I] in the last sentence of the definition on page 1. On page 2, R to R should read R -> R, and (alpha(c)-alpha(b))+(alpha(a)-alpha(c)) should read (alpha(b)-alpha(c))+(alpha(c)-alpha(a)). On page 6, the last line should read f(x_0)(y-x_0) instead of f(x_0)(x-x_0). In the first paragraph on page 8, "assumed to be Riemann integral" should read "assumed to be Riemann integrable". On pages 12 and 13, there are three instances of "for all x in I" missing in the proof of Theorem 5. On page 16, all occurrences of I should be [a,b], and on the second line "minorizes" should just read "less than or equal to". On page 17, the partition Q does not refer to the rational numbers.
• (Mar 13) I found three missing right parentheses on page 13.
• (Mar 12) Alex pointed out that in the first bullet of page 15, "Theorem 8(h)" should read "Theorem 8(h) from Week 9 notes".
• (Mar 12) Updowncharm noted that the words "are also Riemann integrable" were missing from the end of the statement of Theorem 3.

• (Mar 17) Marion suggested that the analogy between upper Riemann integral/lower Riemann integral/Riemann integral and lim sup/lim inf/lim be clarified.
• (Mar 13) Jared noted that in the definition of common refinement on page 6, the interval I is used for two different purposes similarly, and one of the two usages should be relabeled. For instance, one can define P # P' to equal { J intersect K: J in P and K in P' }.
• (Mar 13) Marion noted that on the definition of piecewise constant integral on page 8, "for each J" should read "for each J in P".
• (Mar 11) Alex pointed out on the first paragraph on page 5 that "one of the intervals K in P contains P" should read "one of the intervals K in P contains b", and "Since K contained in I" should read "Since K is contained in I".
• (Mar 10) Nameless pointed out that in the example of a common refinement on page 6, the point 2 is duplicated in the partition, and [2,3) should instead read (2,3). Conversely, in the example on page 7, there is a {2} missing from the second partition that f is piecewise constant over.
• (Mar 8) Evan noted that Lemma 9 was not proven in these notes or assigned as homework; I have added the proof to the notes. Also, in the definition of finer partition at the bottom of page 5, J should be a subset of K, not of P.
• (Mar 7) I noticed that the word "Let" was missing at the beginning of the second definition of page 11 (the upper and lower Riemann integrals).
• (Mar 4) I noticed that "Weeks 9 notes" should read "Week 9 notes". Also, on the last equation on page 15, sup should read inf. Finally, in the discussion on the final page concerning the Riemann-Stieltjes integral, I changed alpha(J) to alpha[J] for compatibilty with Week 10 notes.`
• (Mar 3) Twistorstrings noted that all occurences of "p.c." in Theorem 13 on page 14 should be deleted. Also, in the remark following on page 15, "Theorem 13(e)" should be "Lemma 10".
• (Mar 1) Twistorstrings pointed out that in Theorem 8(g) on page 10, the integrals should be piecewise constant integrals (i.e. the prefix p.c. is missing). On page 11, in the definition of upper Riemann integral, the sup should be an inf. (This change should already be fixed on your printed copy of the notes).

• (Mar 7) Tal pointed out that in the second bullet of page 30, "lemma states" should read "proposition states".
• (Mar 3) Nameless pointed out that on page 18, in the paragraph before "Example", "automatically be defined" should read "automatically be undefined". In Theorem 15(c) on page 22, f(x_0) and g(x_0) should read f'(x_0) and g'(x_0) (this was also pointed out by Natasha and Marion). In the definition on page 24, the second occurrence of "local maximum" should instead read "local minimum"; in the inverse function theorem on page 28, f^{-1} should be assumed continuous at y_0, not at x_0; in Proposition 25 on page 30, the phrase "there exists" should be deleted from "then there exists g'(x_0) = 0".
• (Feb 27) Tal pointed out that in the definition on page 2, M should be specified as a real number and not just as a number (the point is that M is explicitly prohibited from being +infinity or -infinity).
• (Feb 26) I found out that I did not state where Corollary 19 (the mean value theorem) was proven (it's proven in HW).
• (Feb 26) Tal pointed out on page 10, in the last sentence of the second bullet (involving the function f(x) = 1/x), that "f(x) 0.1-close" should read "f(x) 0.1-close to f(x_0)". Also, on page 11, in the definition of equivalent sequences, all sequences should start at n=m, not n=0, and the constraint "n in N" should be replaced by "n >= m".
• (Feb 24) Anonymous_1 pointed out that "Math 135" should instead read "Math 136" on page 6.
• (Feb 24) Jared pointed out that "at least on square root of 2" should read "at least one square root of 2" on page 7. After the definition of uniform continuity on page 11, "whenever x in X is delta-close to X" should read "whenever x in X is uniformly close to x_0". In the definition on page 16, the terms "converges to" should be italicized (to represent a new term being defined). On page 19, in the first bulleted paragraph, the first parenthetical comment is ended at the incorrect place - it should end just before "then g is differentiable".
• (Feb 23) Marion pointed out that the parenthetical remark on the definition in page 3 was grammatically incoherent (it should state that the value of f at x_0 is greater than or equal to the value of f at all other points in X).
• (Feb 21) Tal observed on page 1 that "f(x_n) also converges f(x_0)" should read "(f(x_n))_{n=1}^infinity also converges to f(x_0)".
• (Feb 20) I found out that in the definition of uniform continuity on page 11, "two points in x are" should read "two points in X which are".
• (Feb 18) Twistorstrings pointed out on page 3, in the definition of minimum, f(x_0) < = f(x_0) should instead read f(x_0) < = f(x) (this was also pointed out by Evan). Also, on page 7, the function x^3 + x should be x^3 - x, and the numbers +10 and -10 should instead be +6 and -6. Finally, on page 8, the definition of strictly monotone increasing should read f(y) > f(x) instead of f(y) < f(x).

• (Feb 20) Marion observed that on page 15 (or 16, in some versions) in the first example for continuous functions, the limits should read "x -> x_0" rather than "x_0 -> x" or "x_0 is an element of x".
• (Feb 19) Marion pointed out that on the top of page 3, "last week's notes" should instead refer to "Week 3/4 notes" (when referring to limit points).
• (Feb 18) Tal pointed out that the "If... then" in Proposition 2(a) should be phrased using a comma instead of a period. In the statement of Lemma 5, the grammar was strange in several places (adherent point instead of an adherent point, converge instead of converges). Also "a adherent" should be "an adherent" at multiple places in the notes. On the second example on page 10, both occurrences of "0.1-close to 2" should instead read "0.1-close to 4", and the example is also missing a right-parenthesis.
• (Feb 18) Marion observed that a right parenthesis was missing in the definition of subsequence on page 1.
• (Feb 18) Nameless pointed out that in the discussion on dummy variables on page 12, y_0 should be x_0. In the informal discussion of Proposition 8 on page 13, the final limit involving g(x) should be "x -> x_0; x in E" instead of just "x -> x_0". In the first example on Page 15, "g: R to R" should instead read "g: R --> R". Finally, the Feb 13 correction by Twistorstrings contained an error (now fixed).
• (Feb 18) Twistorstrings and Nameless both pointed out that in page 12, both in the Definition and in Proposition 6, "Let L be a number" should be "Let L be a real number".
• (Feb 17) Evan and Anonymous_1 pointed out that in the informal explanation on Page 15, the limit should be as x -> x_0, not "x is an element of x_0".
• (Feb 16) Marion and Nameless pointed out that the expression f+g(x) on page 7 should be written as (f+g)(x) to avoid confusion.
• (Feb 13) Twistorstrings noted that the definition of closure just before Lemma 4 should state that the closure of X is the set of all adherent points of X, not the set of all adherent points of R.
• (Feb 12) Updowncharm pointed out that on the first definition in page 8, "y is an element of E" should read "y is an element of X", and in Lemma 5 on page 9, "x is adherent to R" should read "x is adherent to X".
• (Feb 11) Evan pointed out that on Page 6, in the second paragraph, "x is not equal to X" should read "x is not an element of X".
• (Feb 11) Evan pointed out that on page 1, the first sequence is missing the term 0.001 between 1.001 and 1.0001. Also, on page 3, it is not mentioned that the proof of Proposition 3 is deferred until the homework.

• (Feb 23) Marion pointed out that the second part (a) in the Ratio Test (Corollary 28 on page 33) should be (b).
• (Feb 17) Tal noted that on page 34 (or 35 in some versions), in the third bullet in the proof of Lemma 30, a right parenthesis is missing after "since x>1".
• (Feb 13) Jared pointed out that I reported Marion's Feb 6 and Nameless's Feb 11 corrections below incorrectly; this is now fixed.
• (Feb 12) Marion pointed out that in Proposition 9, h should be a bijection with range X (rather than with unspecified range).
• (Feb 11) Tal noted that all references to Lemma 1(a)-1(g) on page 3 should instead refer to Lemma 2(a)-2(g).
• (Feb 11) Nameless pointed out that in Proposition 2(e) on page 2, one should use a variable other than n (since n was fixed at the beginning of the Proposition). For instance, one should say something like "x^{1/k} is a decreasing function of k", etc.
• (Feb 10) Nameless pointed out that in page 15 (Fubini for finite sums) the third term in the equality should sum over (y,x) in Y x X, not (y,x) in X x Y.
• (Feb 6) Marion noted that in Lemma 8(f) on page 9, "a_i > = b_i" should instead read "a_i < = b_i".
• (Feb 6) I observed that in Proposition 15, it was not mentioned that the proof of this proposition is part of the Week 5 homework.
• (Feb 5) Evan noted that in the middle of page 35, the equality |x^{q_n} - x^{q_m}| = x^M (x^{q_n - q_m} - 1) should instead read inequality |x^{q_n} - x^{q_m} < = x^M (x^{q_n - q_m} - 1).
• (Feb 4) Evan noted that on the bottom of page 29, both occurences of "inequality" should read "equality".
• (Feb 4) Evan noted that on page 22, S_N should be the sum of a_n from 1 to N, not from 1 to infinity.
• (Feb 4) Tal noted that the notes should be titled "Week 5" instead of "Weeks 5".
• (Feb 1) I discovered that Proposition 7 is a duplicate of Theorem 30 from last week's notes.

• (Jan 30) I found out in Theorem 30 that Proposition 27(d) should read Proposition 27(f).
• (Jan 29) Alex noted a missing parenthesis after "Thus a_0 is the smallest element of X" on page 4 of week 3/4 notes.
• (Jan 29) Jared pointed out that in Proposition 27(f) on page 32 of Week 3/4 notes, "converges c" should read "converges to c".
• (Jan 28) Twistorstrings pointed out that in Proposition 24 on page 26 of Week 3/4 notes, the sequences should start at n=m, not n=1.
• (Jan 28) Jared pointed out that "the supremum of this sequence should be 1" should read "the supremum of this sequence should be -1" in page 30 of the Week 3/4 notes.
• (Jan 27) Tal pointed out that "a incomplete sketch" in page 4 of Week 3/4 notes should read "an incomplete sketch".
• (Jan 27) Jared pointed out that on page 25 of Week 3/4 notes, "Suppose that M is an lower bound" should be "Suppose that M is a lower bound". Also, in Propositions 24 and 25, "sup_{n=1}^{infinity} a_n" should instead read "sup (a_n)_{n=1}^{infinity}".
• (Jan 27) Anonymous_1 pointed out that I did not define the notion of equality in a Cartesian product, and that this is important (e.g. for Q8 of Assignment 3). Two pairs (x,y) and (x',y') in X x Y are considered equal iff one has both x=x' AND y=y'. Thus for instance (2,3) and (2,3) are equal, but (2,3) and (2,4) are not (and (2,3) and (3,2) are definitely not equal).
• (Jan 27) Jared pointed out that the parenthetical remark near the bottom of page 19 of Week 3/4 notes should be in the past tense ("was" instead of "is"), and that "x+y" should read instead "xy" in Theorem 21(b) on page 20.
• (Jan 25) Jennifer pointed out that in the proof of Lemma 8 on page 7 of Week 3/4 notes (second equation from the bottom), f(n',m') should equal a_{n'} + m', not a_{n'} + m.
• (Jan 24) Nameless pointed out the following corrections in the Week 3/4 notes. On page 15, after the second definition, "The starting index N" should read "The starting index m". In the third definition, meanwhile, "a_n and a_m are eps-close for all n >= N" should read "a_j and a_k are eps-close for all j,k >= N". In the definition on page 16, "a_n is close to L" should read "a_n is epsilon-close to L". In the proof on page 17, c and c' should read L and L' respectively, while in the statement of that proof "to converges" should be "to converge". In Theorem 21(d) on page 20, x+y should read x-y.

• (Feb 11) Nameless pointed out that the laws of exponentiation (Propositions 3 and 4) were only stated for rational bases, but actually we need to extend those results to the case where the bases are real. The point is that once we have proven that all the usual laws of algebra and order hold not only for the rationals, but also for the reals, Propositions 3 and 4 also automatically hold for real bases.
• (Feb 5) Anonymous_1 pointed out that the definition of a sequence was never rigorously defined. Rigorously speaking, a sequence (a_n)_{n=m}^infinity of rational numbers is a function from the set {n in Z: n > = m} to Q, i.e. to each integer n greater than or equal to m, we assign some rational number a_n. A sequence of real numbers is exactly the same, except that Q is replaced by R, of course.
• (Jan 31) Alex noted a missing right parenthesis near the bottom of page 23.
• (Jan 26) Anonymous_1 pointed out it was unclear what the cardinality of the empty set was in the Week 2 notes. I have now clarified: the empty set has equal cardinality with the set {i in N: 1 < = i < = 0} (and also equal cardinality with {i in N: i < 0}) and hence has cardinality 0. In particular, the empty set is finite, and not infinite (since 0 is a natural number).
• (Jan 25) Jennifer pointed out that in the proof of Lemma 31 of Week 2 notes, "integer" would be better as "natural number". Some other corrections in that area; one of the N's should be N (since it doesn't make sense for i to be an element of N, since N is a number rather than a set). Also, for compatibility with lecture I added the remark that one can use either {i in N: 1 < = i < = N} or {i in N: i < N} when defining what it means for a set to have N elements, since these two sets are bijective.
• (Jan 24) Alex pointed out a missing parenthesis at the bottom of page 25 in Week 2 notes (after "... lie between -M and M").
• (Jan 21) I emphasized that the proof of the least upper bound property in the Week 2 notes is optional reading; the result is definitely important, but the proof is somewhat long and tedious, and you should probably focus more on other aspects of the course for now. (Incidentally, in the first paragraph of that proof, "at least one upper bound" should read, somewhat confusingly, "at least one least upper bound").
• (Jan 20) Nameless pointed out in the Week 2 notes that in the statement of Proposition 2(h), "delta |w|" should read "delta |x|". In the proof of Proposition 21 on page 28, the phrase "n != 1, by hypothesis" should read instead "n >= 1, by hypothesis". Similarly on page 34, in the proof of Proposition 28, "and hence y != E" should read "and hence y is not an element of E". Finally, on page 36, "we match the set we are trying to set" should be "we match the set we are trying to count".
• (Jan 20) Tal pointed out some typos in Week 2 notes. In Page 7, Proposition 4(b) is either meaningless (since 0^n is not defined for negative n) or subsumed in Proposition 3(b), so I deleted it; similarly Proposition 4(c) should only be used when x and y are positive (so I replaced all occurrences of ">= 0" with ">0"). On page 14, the square root of 2 should be "1.41421..." instead of "1.414121..." (and pi should be "3.14159..." instead of "3.14156" in page 2 of the decimal handout). Near the bottom of page 21, "To do this we a definition" should read "To do this we first need a definition", and "there exists an c>0" should read "there exists a c>0" (similarly at the beginning of the proof on page 23).
• (Jan 20) Twistorstrings noted that Proposition 19 of Week 2 notes is not actually proven in the notes, nor is it assigned as homework. This problem was initally intended to be the second part of Q8 (as it follows quickly from Proposition 18) but got omitted. So I have sketched instead in the notes how Proposition 19 follows from Proposition 18 and a little algebra.
• (Jan 19) Nameless pointed out that in Week 2 notes, "z and w are close" should read "z and w are delta-close" near the bottom of page 5; that "p := q" should be "p' := q" near the bottom of page 8; that a right parenthesis was missing after "1,1/2,1/3,1/4..." in the second example on page 11; that "M greater than or equal to r" should read "M greater than or equal to 4" on page 13; that "Because of this definition" should read "Because of this Proposition"; and that "epsilon-close" and "epsilon/2-close" should read instead "epsilon-steady" and "epsilon/2-steady" in the middle of page 22.
• (Jan 18) Jennifer pointed out that b := w-x should read b := w-z in the bottom of page 5 of Week 2 notes.
• (Jan 17) I discovered that "unmber" in Lemma 11 of Week 2 notes should read "number".
• (Jan 17) Marion pointed out that the numbering of the components of Proposition 2 of Week 2 notes was off (part (f) was repeated twice). In particular, 2(g) should really be labeled 2(h), with Q2 of Homework 2 changed accordingly. Marion also observed that in the proof of Proposition 2(h), (x+a)(w+z) should read instead (x+a)(z+b).
• (Jan 16) Jennifer pointed out that 3+5/n should read 3+4/n in the example on page 17 of Week 2 notes.
• (Jan 14) Evan pointed out that in the definition of upper bound on page 30 of Week 2 notes, the phrase "for all x in M" should read "for all x in E".
• (Jan 13) Twistorstrings noted that the phrase "If x=y, then" should be inserted at the beginning of Proposition 2(a) on page 5 of Week 2 notes. Also, in the first definition on page 14 of Week 2 notes, the second sequence should read (b_n)_{n=0}^infinity, not (b_n,b_2,... Finally, the notion of decreasing function on page 15 of Week 2 notes has to be defined (a sequence a_1, a_2, a_3, ... is decreasing if one has a_m < a_n whenever m > n, or equivalently if a_{n++} < a_n for all n; one can show the two definitions are equivalent using induction).

• (Jan 14) Nameless pointed out that the last Jan 10 correction was reported incorrectly (Page 23 should have been Page 26); this is now fixed. Also, in the definition of addition of rationals (on or near Page 34 of Week 1 notes), the denominator bd should be placed in parentheses (bd).
• (Jan 14) Jared pointed out that x+y=x+y on page 30 of Week 1 notes should read x+y=y+x. (A similar error occurs on Page 36).
• (Jan 13) Marion noted that "Propositions 1,2,3" on Page 23 of Week 1 notes should read "Lemmas 1,2 and Proposition 3", and also observed that the Week 1 notes should at some point (also near page 23) mention that one uses the usual notational convention of letting multiplication take precedence over addition (i.e. ab+c equals (a * b) + c, not a * (b+c)). Similarly for the other arithmetic operations.
• (Jan 12) Tal observed that the Week 1 notes were a little unclear as to how the natural numbers were defined. What should have been emphasized is that we define the natural numbers to be the elements of a certain number system N. We then assume that this number system N obeys certain axioms, specifically Axioms I-V from the Week 1 notes. Thus it is hypothetically possible that the definition of natural numbers in these notes does not correspond to our "intuitive" notion of natural numbers (for instance, we entertained the hypothetical possibility that 0.5 was an element of N and thus qualified as a natural number). However, the purpose of the Axioms I-V is to rule out such unintuitive behavior, and ensure that the number system we have defined does correspond well with our intuitive notion of the natural numbers. I've changed the presentation in the Week 1 notes to reflect this.
• (Jan 11) Andrew noted that the equation "a+e = c+f" on page 27 of Week 1 notes should have read "a+f = b+e". Also, the equation "c+e = d+f" appearing a little earlier should have been "c+f = d+e".
• (Jan 11) Nameless noted on page 15 of the Week 1 notes that the statement "0.5 cannot be a half-integer" would be better phrased as "0.5 cannot be a natural number". Also some minor typographical errors in that paragraph have been fixed.
• (Jan 11) Nameless pointed out near the bottom of page 20 of Week 1 notes that "By definition of induction" should read "By definition of addition".
• (Jan 11) Nameless observed a right parenthesis was missing in the definition of positive and negative rationals on page 36 of Week 1 notes.
• (Jan 11) Updowncharm observed that the distributive law (Proposition 10) should be appear _earlier_ than the associative law in pages 23-24 of the Week 1 notes, instead of appearing two paragraphs after it, to avoid the impression of circularity.
• (Jan 10) Twistorstrings noted an incomplete sentence before the last definition in page 26 of Week 1 notes (this sentence should end "...very helpful in later weeks").
• (Jan 8) Moomin noted that in the second equation on page 6 of Week 1 notes, dx dy should read dy dx.
• (Jan 8) Kwo-Ling pointed out a hidden assumption in the Week 1 notes: that the sum (or product) of any two natural numbers is again a natural number.  However, this is easily proven using induction and Axioms I and II.
• (Jan 7) Anonymous_1 pointed out that in the definition of multiplication in page 23 of the week 1 notes, two occurrences of the word "add" should instead read "multiply".
• (Jan 6) Kwo-Ling pointed out that in the proof of Proposition 5 in page 21 of the Week 1 notes, Axiom III should read Axiom IV.

• (Mar 19) Natasa and Stefan noted that the definition of closure in the final was incorrect; Natasa also pointed out the definition for upper Riemann integral was also incorrect.
• (Mar 4) Marion pointed out that the date of one of the announcements was incorrect; this is now fixed.
• (Mar 4) Jared pointed out on the solution to Q3 of the midterm that "S - epsilon is less than E" should read "S - epsilon is less than S".
• (Mar 2) Tal discovered a typo in the announcement concerning the second midterm, which is now fixed.
• (Feb 27) Nameless and Twistorstrings both discovered a duplicate answer in the continuity Java quiz, which has now been fixed.
• (Feb 10) Nameless pointed out that in the inequalities Java quiz question involving a^2, many of the a^2s in the answers were incorrectly written as a's.
• (Feb 4) Tal noted that in many statements in the notes, variables were used without declaring what type they were (real number, integer, etc.) The default convention is this: unless specified otherwise, all variables using the letters a,b,c,x,y should be real; all variables using the letters n,m,i,j should be natural numbers or integers. For other variables, the conventions vary. The letters f,g,h usually denote functions. Letters like A,B,C,X,Y,Z often denote sets. epsilon and delta usually denote positive, but small, real numbers. The letters p,q,r can denote real numbers, rationals, integers, or natural numbers. In most cases one can work out what the types of the variables are from context.
• (Feb 4) Jared noted that in the course handout at the beginning of this course, "an UCLA account" should read "a UCLA account".
• (Jan 28) Jared noted some grammar errors in the countability Java quiz "a uncountable" and "an countable" instead of "an uncountable" and "a countable", and that a +infinity and -infinity were switched in one of the answers in the sequences quiz, and "(a_n)_{n=1}^infinity be a Cauchy sequence" should have read "(a_n)_{n=1}^infinity to be a Cauchy sequence".
• (Jan 27) Tal suggested that one clarify the difference between a function and a bijection. A function f: X -> Y is a mapping which assigns exactly one output f(x) (in Y) to each input x (in X); a bijection f: X -> Y is a function such that every output y (in Y) comes from exactly one input x (in X), i.e. for every y in Y there is exactly one x in X such that f(x) = y. The two notions should not be confused.
• (Jan 22) Twistorstrings observed that the use of the word "if" in the Definitions was somewhat ambiguous. In all the notes, when I define a property P using a Definition of the form "Definition. An object x has property P if Y happens", what I mean by this is that if Y happens, then x has property P by definition, and conversely if x has property P, then Y happens. Thus the "if" might be more accurately rendered as an "if and only if" (sometimes abbreviated "iff"). Thus I have changed all the "if"s to "iff"s in the definitions for clarity.
• (Jan 22) Alex found some bad links in the official course description page; I've contacted the UCLA math department webmaster to have them fixed.
• (Jan 21) Twistorstrings pointed out that the link to the class web page from the syllabus web page was incorrect (it was pointing to an old 132 class page of mine instead). This is now fixed.
• (Jan 20) Alex pointed out that in page 2 of the decimal system handout, "Let 0_1, ..., x_n" should read "Let x_0, x_1, ..., x_n"; that in the first example of page 30 of week 2 notes, a right parenthesis ) is missing (and the parenthetical comment could be clarified more); and that the parentheses () are optional in sup(E) and inf(E) (thus sup E and inf E are also valid mathematical expressions).
• (Jan 19) Nameless observed that the statement "f(x) in f(S) <==> x in S" in page 6 of the set theory handout is incorrect; it is true that x in S implies that f(x) is in f(S), but not conversely (e.g. if f(x) = x^2 and S = {-1,0,1,2}, then f(-2) is in f(S) but -2 is not in S). The correct statement should be that "y is in f(S) if and only if y is equal to f(x) for some x in S".
• (Jan 12) Anonymous_1 noted that the permissions on the decimal system handout were not set correctly; this is now fixed.
• (Jan 10) Twistorstrings noted a technical error in the Java Quiz (it required Sun Java VM 1.4.1 but was downloading the Java VM 1.1.1 plugin instead).
• (Jan 9) Twistorstrings noted that the statement x^2 = 5 at the bottom of page 11 of the logic handout should read x^2 = 4 instead.
• (Jan 8) Moomin noted that the weighting for the final should be 40%, not 45% (before the additional 5% was taken into account), and that the discussion was mistakenly printed as MS 5127 instead of MS 5148.
• (Jan 8) Jared discovered one of the links on the math department Math 131 web page was incorrect; this has been fixed.
• (Jan 7) Nameless pointed out an error in the vector space java quiz (now fixed).
• (Jan 6) Arthur pointed out that the date for the first midterm was incorrect; this has now been fixed (the correct date being Friday, Jan 31).

• It is possible for students to earn bonus points to improve their grade.
• Each point adds 0.1% to the final grade, however at most 20 bonus points per student may be applied to the grade (i.e. it is possible to earn more than 20 bonus points, but after 20 there will be no further effect on your grade). Thus the theoretical maximum score for the class is 102%.
• Bonus points can still be awarded after Wednesday, March 19 (the day of the final), but they will no longer count toward the grade (i.e. you cannot, after receiving your grade, try to suddenly earn a large number of bonus points to try to alter it).
• Excess bonus points will not be carried forward into the 131BH course next quarter, since not every student in that class will come from this 131AH one. Bonus points cannot be transferred or otherwise manipulated.
• The bonus point system is designed to encourage class participation, but should not be viewed as a way to salvage a bad grade or as a substitute for doing homework or reading the textbook - this is why your bonus grade is limited to 2%!
• You can earn a bonus point whenever you discover an error (or have a good suggestion to make) in any of my printed notes, homework, exams, solutions, class web page material, java applet questions, or anything I say in Virtual Office Hours.
• Errors can be typographical (i.e. spelling or grammar errors), factual (e.g. concerning the date of the final), mathematical, logical, or occasionally even aesthetic. Basically, the rule is that if it warrants a change to the printed notes (or other web page material), it deserves a bonus point.
• A group of related errors will be considered as a single error for the purposes of awarding bonus points. (This is to avoid a single, widely spread, error overwhelming the system, or to avoid a single error being "sliced" into many smaller errors in order to stretch the bonus point value).
• Detection of verbal mis-steps, for instance in lecture or office hours, will not earn bonus points, due to unavailability of a reliable written record.
• To qualify for a bonus point you must notify me of the error, either by e-mail or in Virtual Office Hours (you can use your exam nickname instead of your real name in Virtual Office Hours if you wish).
• Once an error is pointed out by one student (or by myself), it cannot be re-used by another student to earn bonus points (unless I judge the error notifications to be independent and nearly simultaneous).
• You can also earn a bonus point by correctly answering any question posed by another student in the Virtual Office Hours (as long as the question has some relationship to the course).  Again, once an answer is used by one student to earn a bonus point, it cannot be re-used by another student (unless he or she comes up with a very different, but still correct, answer).
• You may also be able to earn a bonus point for asking a particularly good question in Virtual Office Hours.  (Questions such as "How do you do Question 4 of Homework 2?" or "Will Section 4.3 be on the final?" are generally not considered good questions for this purpose).
• Last, but not least, you can earn bonus points by suggesting a new question for the Java quiz. Each question suggested is worth one bonus point. To qualify, each question must be original (i.e. not a minor modification of an existing question), relevant to the course, educational (i.e. the correct answer should teach you something about the concepts covered by the question) and should come with at least two incorrect but plausible answers. Contributors will be acknowledged in the data files of the java quiz as well as on the bonus point system.
• Any attempt to abuse the bonus point system may result in forfeiture of all bonus points by the abuser.
• These rules and regulations are subject to change without notice.