Negation Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium

Build the truth table for

\neg p

and verify that

p \land \neg p

is always false (contradiction) and

p \lor \neg p

is always true (tautology).

Solution

1
For $p = T$ : $\neg p = F$ . Then $p \land \neg p = T \land F = F$ and $p \lor \neg p = T \lor F = T$ .
2
For $p = F$ : $\neg p = T$ . Then $p \land \neg p = F \land T = F$ and $p \lor \neg p = F \lor T = T$ .
3
In both rows, $p \land \neg p = F$ and $p \lor \neg p = T$ . Confirmed.

Answer

p \land \neg p \equiv \text{False (contradiction)},\quad p \lor \neg p \equiv \text{True (tautology)}

A statement and its negation cannot both be true simultaneously (law of non-contradiction), and one of them must be true (law of excluded middle).

About Negation

The negation of a statement $P$ , written $\neg P$ , is the statement with the opposite truth value: true when $P$ is false, and false when $P$ is true.

Learn more about Negation →

More Negation Examples

Example 1 easy

Write the negation of each statement and determine its truth value: (a) '[formula]', (b) 'All cats a

Example 2 medium

Simplify [formula] using logical laws.

Example 3 easy

Write the negation of: (a) 'It is raining', (b) '[formula]', (c) 'Every student passed.'

← All Negation Examples Negation Concept

Related Concepts

Logical Statement