Logic and Mathematical Statements

If...then... statements

In general, a mathematical statement consists of two parts: the hypothesis or assumptions, and the conclusion. Most mathematical statements you will see in first year courses have the form "If A, then B" or "A implies B" or "A $\Rightarrow$ B". The conditions that make up "A" are the assumptions we make, and the conditions that make up "B" are the conclusion.

If we are going to prove that the statement "If A, then B" is true, we would need to start by making the assumptions "A" and then doing some work to conclude that "B" must also hold.

If we want to apply a statement of the form "If A, then B", then we need to make sure that the conditions "A" are met, before we jump to the conclusion "B."

For example, if you want to apply the statement "$n$ is even $\Rightarrow$ $\frac{n}{2}$ is an integer", then you need to verify that $n$ is even, before you conclude that $\frac{n}{2}$ is an integer.

In mathematics you will often encounter statements of the form "A if and only if B" or "A $\Leftrightarrow$ B". These statements are really two "if/then" statements. The statement "A if and only if B" is equivalent to the statements "If A, then B" and "If B, then A." Another way to think of this sort of statement is as an equivalence between the statements A and B: whenever A holds, B holds, and whenever B hold, A holds.

Consider the following example: "$n$ is even $\Leftrightarrow \frac{n}{2}$ is an integer". Here the statement A is "$n$ is even" and the statement B is "$\frac{n}{2}$ is an integer." If we think about what it means to be even (namely that n is a multiple of 2), we see quite easily that these two statements are equivalent: If $n=2k$ is even, then $\frac{n}{2} = \frac{2k}{2} = k$ is an integer, and if $\frac{n}{2} = k$ is an integer, then $n=2k$ so $n$ is even.

In everyday use, a statement of the form "If A, then B", sometimes means "A if and only if B." For example, when most people say "If you lend me \$30, then I'll do your chores this week" they typically mean "I'll do your chores if and only if you lend me \$30." In particular, if you don't lend the \$30, they won't be doing your chores.

In mathematics, the statement "A implies B" is very different from "A if and only if B." Consider the following example: Let A be the statement "$n$ is an integer" and B be the statement "$\frac{n}{3}$ is a rational number." The statement "A implies B" is the statement "If $n$ is an integer, then $\frac{n}{3}$ is a rational number." This statement is true. However, the statement "A if and only if B" is the statement "$n$ is an integer if and only if $\frac{n}{3}$ is a rational number," which is false.

Mini-Lecture.

Below is a mini lecture about if-then statements.

Example.

Consider the statement "Suppose that it's raining. Then there is a cloud in the sky.".

(i)	Determine the hypotheses/assumptions and the conclusion.
(ii)	Rewrite this statement explicitly in the form "If A, then B" using Part (i).
(iii)	Is this statement true or false?

Solution.

(i)	The hypothesis we are making is that it is raining. The conclusion we are making is that there must be a cloud in the sky.
(ii)	"If it's raining, then there must be a cloud in the sky."
(iii)	This statement is true. (Based on all that is currently known about how rain works!)

Example. Consider the statement "$x > 0 \Rightarrow x+1>0$". Is this statement true or false?

Solution. To determine it's truth value, first we look at the hypothesis: $x>0$. Whatever we want to conclude, it is a consequence of the fact that $x$ is positive.

Next, we look at the conclusion: $x+1>0$. This statement must be true, since $x+1 > x > 0$.

This means that the statement is true.

Example. Consider the statement "If $x$ is a positive integer or a solution to $x+3>4$, then $x>0$ and $x> \frac{1}{2}$." Is this statement true?

Solution. To determine if it's true, let's look first at the assumptions. We are assuming that either $x$ is a positive integer, or that it solves the inequality $x+3>4$.

Next let's consider the conclusion. We are concluding that $x$ must satisfy both inequalities $x>0$ and $x > \frac{1}{2}$. If we look more closely, we see that once we satisfy the second inequality, the first is redundant. (If $x>\frac{1}{2}$, then it must already be larger than zero.)

Now, in order for this statement to be true, we need that if $x$ solves either of the assumptions, then it must solve $x>\frac{1}{2}$. Well, the first assumption is that $x$ is a positive integer, which means that $x\geq 1$, so in this case the conclusion holds. The second assumption is that $x+3>4$, or equivalently, that $x>1$, which means the conclusion holds as well.

Example. Consider the statement "$0>1 \Rightarrow \sin x = 2$". Is this statement true or false?

Solution. To determine it's truth value, first we look at the hypothesis: $0>1$. This is obviously false!

So the statement is true! (Why?) This kind of statements "A $\Rightarrow$ B" where A is false are called vaccuously true.

A statement "A $\Rightarrow$ B" is true when the relation "A implies B" is true, not when A, or B, or A and B are true. It states that "if A is true, then B must also be true".

This means that when A is false, the statement doesn't conclude anything.

So whenever the hypothesis A is false, a statement "A $\Rightarrow$ B" is always true! (independently of whether B is true or false)