0. ``For all'' and ``there exists''

In mathematics, many assertions involve the idea of ``for all''. Others involve the idea of ``there exists''. Still others have both as ingredients.

``For all $x$'' has the shorthand form $(\forall x)$.

``There exists $x$'' has the shorthand form $(\exists x)$.

There are many ways to express these two concepts in English but you can learn to recognize them.

Here are some examples of statements that can be re-expressed in a shorter form using these ideas. They all apply to real numbers, which we won't mention explicitly.

Example (i): ``For any $y \geq 0$ there is an $x$ whose square is $y$'' becomes
$(\forall y \geq 0)(\exists x)(y = x^2)$.

Example (ii): ``Any nonnegative real number is in the range of the squaring function $y = x^2$.'' becomes the same as (i).

Example (iii): ``Given any $y$ with $\vert y\vert \leq 1$ there is an $x$ so that $(x,y)$ lies on the circle $x^2 + y^2 = 1$ '' becomes
$(\forall y)(\vert y\vert\leq 1 \Rightarrow (\exists x)(x^2 + y^2 = 1))$.

The ingredients in these mathematical ``sentences'' (assertions) are $\forall$, $\exists$, variable letters, ``and'', ``or'', ``not'', $=$, inequalities, $\Rightarrow$, and parentheses. If we were talking about sets we would use $\in$ also. Some of these ingredients can be re-expressed using others, but it's easier just to use them all. For ``not $=$'' you can use $\neq$, etc.

Problem Q-1. Write each of these in shorter form as in the examples above.

(a) The range of the function $y = x^3 - x$ is all numbers.

(b) The real numbers obey the law $x+y = y+x$.

(c) If a number is positive so is its square.

(d) There is a solution to the equation $1 -8x + x^2 = 5$.

(e) It is possible to find $x$ and $y$ so that $17x + 25y = 6$ and $101 x - 37 y = 13$.

(f) For at least one $a$, the equation $ax = x+1$ has no solution. (For ``has no'' you can use $\not \exists x$.)

(g) Re-do (f) using the idea that ``there does not exist $x$ with property $P$'' is the same as as ``for all $x$ we have not-$P$''. (So it isn't ever really necessary to use $\not \exists$.)

(You are not asked to prove any statements or solve for any answers; just restate them.)