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Negation

As we explore logic problems in the realm of mathematics, we often learn new ways of saying things that would seem like common sense in other contexts. For instance, negation may be defined as the contradiction or denial of something. The negation of a statement means that the statement is not true. For example, the negation of statement $p$ is not $p$ .

In this article, we'll cover what negation looks like in mathematics and how to use it. Let's get started!

Expressing negation mathematically

Mathematicians never want to use words when numbers and symbols would suffice, so you can probably guess that math has a dedicated symbol for negation. It's a tilde (∼) or a negation symbol (¬). The negation of $p$ is not $p$ and is written as ¬ $p$ or $~p$ .

Importantly, $~p$ must be false if statement $p$ is true and true if statement $p$ is false. For instance, let's say that statement $p$ is "5 is a prime number." That means that $~p$ would mean "5 is not a prime number." We know that 5 is a prime number, so statement $p$ is true and therefore $~p$ is false.

We can do something similar with statements that have nothing to do with math. For instance, statement $w$ reads "George Washington is alive." The negation of statement $w$ , $~w$ , would mean that "George Washington is not alive" or more simply "George Washington is dead." In this scenario, we know that Washington passed away in 1799. Therefore, statement $w$ is false and $~w$ is true.

Things can get a little more complex than this. For instance, let's consider statement $r$ , "Bob fought for the United States during World War II". The negation would be "Bob didn't fight for the United States during World War II." While either statement $r$ or $~r$ must be true, we cannot glean any further information. If $r$ is false, that doesn't mean Bob fought for Germany. Maybe Bob wasn't born before World War II. Maybe he fought for another Allied country. It's tempting to conclude that Bob was an enemy of the United States given how the statement is phrased, but that's not how logic works.

Practice questions on negation

a. True or false: if statement $a$ reads "the horse is black", $~a$ would read "the horse is white".

The negation of any statement $a$ is "not $a$ " and we cannot assume any other information. Therefore, $~a$ would read "the horse is not black." That leaves open the possibility of the horse being white, but it could also be red, gray, tan, brown, two-tone, or blue. The answer to this question is "false".

b. If statement $a$ is that $x<7$ and statement $b$ is that $x>7$ , is statement $b$ the negation of statement $a$ ?

This is a tricky one that we'll have to think through logically. At first glance, you might think that $x>7$ would be the negation of $x<7$ because a number that is not less than 7 must be greater than 7. However, you may be overlooking a distinct possibility: what about 7 itself? In this case, $~a$ wouldn't be $x>7$ but rather $x\ge 7$ because we're saying that $x$ isn't less than 7, not that it's greater than anything. Furthermore, either $a$ or $~a$ must be true. If $x=7$ , statements $a$ and $b$ are both false. Therefore, one cannot be the negation of the other.

c. Consider the statement, "Angelica is a software engineer and Tim is a firefighter." Would the negation of this statement be "Angelica is not a software engineer or Tim is not a firefighter"? We're looking at a compound statement this time comprised of two simple statements ( $m$ and $n$ ) connected with an "and." Statement m is "Angelica is a software engineer," so $~m$ would be "Angelica is not a software engineer." Likewise, statement n reads "Tim is a firefighter" so $~n$ is "Tim is not a firefighter." Using DeMorgan's Law, the negation of " $m$ and $n$ " is "(not $m$ ) or (not $n$ )", so the provided statement is indeed the negation of the original statement.

Negation

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