And, Or, Not

And, Or, and Not

Let \(P\) and \(Q\) be statements.

\(P\) and \(Q\) is a new statement that is True if both \(P\) and \(Q\) are True; and false otherwise.

\(P\) or \(Q\) is a new statement that is True if either \(P\) or \(Q\), or both, are True; and false otherwise.

Not \(P\) is a new statement that is True if \(P\) is False, and False if \(P\) is \(Q\).

And

\(P\) and \(Q\) can be written \(P\wedge Q\) (compare with set intersection).

OR

\(P\) or \(Q\) can be written \(P\vee Q\) (compare with set union)

Not

Not \(P\) can be written \(\sim P\), or sometimes \(\neg P\).

Examples

Write the open sentences \(x\not=y\) and \(y\ge x\) as P and Q, P or Q, or not P.

Example

Express the following in the form \(P\wedge Q\), \(P\vee Q\) or \(\sim P\).

\[A\in\{X\in\mathcal{P}(\mathbb{N}):|\overline{X}|<\infty\}\]

Truth Tables

Truth tables are an effective way to keep track of combinations of statements.