Negation

What is negation?

Negation is a unary operator; it only requires one operand. Negation of a proposition is another proposition with the opposite truth value. We use the symbol $\neg p$ to denote the negation of a proposition $p$. Negation is also called the NOT operator.

Examples

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Consider the following propositions:

• $H$: Sara is taking a history course.
• $N$: Sara is not taking a history course.

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$H$ is true only if Sara is taking a history course. $N$ is true only when Sara is not taking a history course. Therefore, $N \equiv \neg H$.

We also denote the negation of a statement $p$ by $\overline{p}$.

Therefore,

$$\neg p \equiv \overline{ p}.$$

Let’s look at another example.

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Consider the following propositions:

• $C$: Charlotte is taller than Emma.
• $\neg C$: Charlotte is not taller than Emma.

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We must be cautious while interpreting the meaning of a statement with a “not” in it. The proposition $\neg C$ says that Charlotte is either shorter than Emma or both have the same height. These subtle differences are significant to notice.

Usually, while translating the negation of a proposition $p$, we put the phrase “it is not the case” before the translation of $p$.

• $\neg C$: It is not the case that Charlotte is taller than Emma.

Typically, we can clarify the meaning of negation by reading the statement carefully in this form.

Truth table

The truth table for the negation operator is given below:

$p$	$\neg p \equiv \overline{p}$
T	F
F	T

As flipping the side of a coin twice brings the same side up. Similarly, applying negation twice does not alter the proposition; that is, $\neg (\neg p) \equiv p$.

In $0/1$-notation, $\neg p$ is defined by the following table:

$p$	$\neg p \equiv \overline{p}$
$1$	$0$
$0$	$1$

In this notation, $\neg p = 1 - p$.

Circuit diagram

We can illustrate the concept of negation through circuit diagrams with electromagnetic switches. Here is the key that will aid you in understanding them:

When $p$ is true, the electromagnet switch is down. This disconnects the electric supply to the bulb. Therefore, the bulb is off, and the value of $\neg p$ is false.

When the $p$ is false, the electromagnetic switch does not get any electric supply. This causes it to go up, and that causes the electric supply to reach the bulb. As the bulb is on, the value of $\neg p$ is true.