1. Ways to say it

Imagine that $P$ and $Q$ are statements, maybe about $x$. Then we can ask whether $P \Rightarrow Q$ and whether $Q \Rightarrow P$.

Problem B-1. ¹ Here are some possible relationships between $P$ and $Q$. Some of them may mean the same as $P \Rightarrow Q$, while some might mean the same as $Q \Rightarrow P$. Decide which is which. Are there any like $Q \Rightarrow P$?

To do this, think about the example above. A simple test is this: $P \Rightarrow Q$ when you can't have $P$ true about some $x$ while $Q$ is false.

(a)	If $P$ then $Q$.
(b)	$P \Rightarrow Q$ ($P$ implies $Q$).
(c)	$Q$ if $P$.
(d)	$Q$ is implied by $P$.
(e)	$Q$ follows from $P$.

(f)	$P$ is a sufficient condition for $Q$.
(g)	$Q$ is a necessary condition for $P$.
(h)	$Q$ is a consequence of $P$.
(i)	$P$ holds only if $Q$ holds.
(j)	not-$Q$ $\Rightarrow$ not-$P$ (the ``contrapositive'').

Problem B-2. Here are some statements about $x$. Write down all the true implications between them, for example, $(1) \Rightarrow (2)$ (really meaning, ``for any $x$, $(1) \Rightarrow (2)$''). For each two, consider both directions.

(1)	$x > 0$
(2)	$x^2 > 0$
(3)	$x \neq 0$
(4)	$x \neq 0$ and ${\frac{\displaystyle 1}{\displaystyle x}} > 0$
(5)	$x = y^2$ for some real number $y$ with $y \neq 0$

Problem B-3. In Problem , find three ``nonimplications'' $P \not \Rightarrow Q$ and for each give a ``counterexample''--in other words, a value of $x$ for which $P$ is true but not $Q$.