3. ``The following statements are equivalent''

Sometimes more than two statements are true or false together. This fact is often expressed as in this example:

The following statements are equivalent:

(1) $x \geq 0$
(2) $x^3 \geq 0$
(3) $x = \vert x\vert$

This would be the same as saying $(1) \Leftrightarrow (2)$, $(1) \Leftrightarrow (3)$, and $(2) \Leftrightarrow (3)$. Often we just say, ``the following are equivalent'', or even just ``TFAE''. To prove such an equivalence, it would be enough to prove that $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (1)$.

Problem B-5. In Problem B-, break the statements into groups so that all the statements within each group are equivalent and no two statements in different groups are equivalent. For each group, state the equivalence using the phrase ``the following statements are equivalent''.

(Don't prove anything. A group could conceivably have just one statement, but that shouldn't happen in this example.)

Note. The moral of this handout is that math is simpler than English! There is really only one underlying mathematical concept here, $\Rightarrow$, but there are a number of ways to express it in English. In this course, be on the lookout for implications.