Logical Implications

This post belongs to my “Intro Math” series, a set of posts designed to give the foundations to advanced mathematics. More info can be found in Intro Math.

Before reading, I recommend Set Theory Part 1 for a brief overview of sets.

Here I’ll be discussing the basic logic found in most mathematical arguments. We’ll be learning about implication, the converse of a statement, contrapositives, and biconditionals.

First, I’ll be using the symbols $p$ and $q$ a lot in this post. These symbols represent arbitrary statements that can either be true or false. Some example statements are “ $x$ is even” or ” $x < 7$ .”

If we want to look at the opposite of a statement $p$ , then we write $\sim p$

Now what does it mean to see the statement ” $p \implies q$ ?” This is a logical sentence found in many theorems, and it can be read as “ $p$ implies $q$ .” It can also be read as “if $p$ , then $q$ .” I think the second sentence is easier to interpret. Basically, if $p \implies q$ , then if $p$ is true, $q$ must be true.

That is the only condition, if $p$ is false, then $q$ can be true or false.

Example 1 – Is the following statement true? If $x \in \mathbb{Z}$ , then $x$ is odd.

No, this is not true. Consider $x = 4$ . Here is a case where $x \in \mathbb{Z}$ and $x$ is not odd. Another wording of this statement is $x \in \mathbb{Z} \implies x$ is odd.

Example 2 – Is the following statement true? If $x \in \mathbb{N}$ , then $x \in \mathbb{Z}$ .

Yes, this is because $\mathbb{N} \subset \mathbb{Z}$ .

Say you have a statement $p \implies q$ . Then the converse of this statement is $q \implies p$ . If $p \implies q$ is true, the converse is not necessarily true. Consider Example 2. While the statement is true, the converse is not.

Say you have a statement $p \implies q$ . Then the contrapositive of this statement is $\sim q \implies \sim p$ . This is always equivalent to saying
$p \implies q$ . A common technique in proving an implication is to prove the contrapositive of it.

Consider the contrapositive of Example 2. $x \notin \mathbb{Z }\implies x \notin \mathbb{N}$ . This is definitely true, and the same as the original statement.

What if the converse of a statement is true? If $p \implies q$ and $q \implies p$ , then we say $p$ and $q$ are equivalent. This is written as $p \iff q$ and called a biconditional. Writing this out gives $p$ is true if and only if $q$ is true. A shorthand is “p iff q” I dislike this version as it could easily be mistaken for a typo.

To prove equivalence of $p$ and $q$ , you must prove $p \implies q$ and $q \implies p$ .

These logical ideas are critical to math and will show up in nearly every theorem. Make sure you understand what they mean for your own learning.

If you’ve read this post and Quantifiers, you have a good basis for understanding symbolic definitions. Test your skills by reading Bijective Functions and working through the definitions provided.

Logical Implications

Published by cfreddy10

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