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>Proofs>What Is the Difference Between Equivalence and Implication?

What Is the Difference Between Equivalence and Implication?

Implication ( $\Leftarrow$ , $\Rightarrow$ ) and equivalence ( $\Leftrightarrow$ ) are two very important logical operators. In this entry you will become familiar with these symbols, and how they work in mathematics.

Logic is the study of correct thinking. It requires sound, valid reasoning, conclusions, and proofs that follow from rules, principles, laws, and terms. Logic has its roots from the time of Aristotle. In mathematics, you use logic as your foundation when working with proofs.

So how is all this connected? Without even thinking about it, you may have already worked with logic quite a bit. Solving an equation is a logical exercise. You look for a solution such that the equal sign becomes true. This is rather exciting. Here are two examples of what implication and equivalence can look like:

When you see an implication arrow ( $\Leftarrow$ , $\Rightarrow$ ), you read it as “if-then”. Such as

If Bono is the lead singer of U2, then he is a member of U2.

If there’s a equivalence sign ( $\Leftrightarrow$ ), you read it as “if and only if”. Such as

$2 x = 4$ if and only if $x = 2$ .

In general, it follows that:

Theory

Implication

p \Rightarrow q

In this case, the statement $q$ is always true, if statement $p$ is true. You say that “ $p$ implies $q$ ” or “if $p$ then $q$ ”.

The implication is valid only one way. This means that $q$ is always true if $p$ is true. But, even though $q$ is true, that doesn’t necessarily mean that $p$ is true.

Example 1

Look at the sentence:

If Lisa lives in London, then she lives the UK.

( $p \Rightarrow q$ , but $n o t$ $p \Leftarrow q$ )

If Lisa lives in the UK, she doesn’t necessarily live in London.

The statement is an implication because the statement is absolutely true only one way. You can’t conclude that since Lisa lives in the UK, she must live in London. She may for instance live in Liverpool.

Example 2

Look at the sentence:

If it rains, the street gets wet.

( $p \Rightarrow q$ , but $n o t$ $p \Leftarrow q$ )

If the street is wet, it isn’t necessarily due to rain.

The street may have gotten wet because someone was watering plants nearby.

Theory

Equivalence

p \Leftrightarrow q

In this case, the statement $q$ is true if $p$ is true ( $p \Rightarrow q$ ), and that the statement $p$ is true if the statement $q$ is true ( $p \Leftarrow q$ ).

You say that $p$ is equivalent to $q$ , since the implication goes both ways. This results in “ $p$ if and only if $q$ ”. Two statements being equivalent to each other means that you have a logical equality.

Example 3

Let’s look at the relationship between David Beckham and his son Brooklyn Beckham.

Brooklyn is David’s son, if and only if David is Brooklyn’s father.

( $p \Leftrightarrow q$ )

Brooklyn being David’s son is equivalent to David being Brooklyn’s father. You may divide this equivalence into two implications:

If Brooklyn is David’s son, then David is Brooklyn’s father.

( $p \Rightarrow q$ )

Thus, the statement Brooklyn is David’s son implies that David is Brooklyn’s father. In addition:

If David is Brooklyn’s father, then Brooklyn is David’s son.

( $p \Leftarrow q$ )

Thus, the statement David is Brooklyn’s father implies that Brooklyn is David’s son.

Since the implication goes both ways, the statements are equivalent.

Example 4

Look at the following statement:

In a triangle, all its angles are equal if and only if all its sides have the same length.

( $p \Leftrightarrow q$ )

This equivalence consists of two implications (here $p =$ “all angles in a triangle are equal” and $q =$ “all sides of a triangle have the same length”):

If all angles in a triangle are equal, then all its sides have the same length.

( $p \Rightarrow q$ )

and

If all sides of a triangle have the same length, then all its angles are equal.

( $p \Leftarrow q$ )

Note that even if the equivalence is true, the two statements themselves don’t have to be true. There are triangles that are not equilateral. What the equivalence says is that if one is true, the other must be true, and vice versa. So if you know that a triangle is equilateral, then you also know that all the angles in it are equal, and if you know that all the angles in a triangle are equal, then all the sides must also have the same length.

Example 5

If $2 x = 4$ , then $x = 2$ . This means that

2 x = 4 \Rightarrow x = 2

But you also know that if $x = 2$ , then $2 x = 4$ . This means that

2 x = 4 \Leftarrow x = 2

Since the implications are true both ways, the statements are equivalent and you write

2 x = 4 \Leftrightarrow x = 2

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