Logical and

Stub grade: Z

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Unimportant as truth tables and the basic logical operators like and, or and not may be assumed

Warning:That stub grade doesn't exist!

Definition

Let [ilmath]A[/ilmath] and [ilmath]B[/ilmath] denote logical statements that are either true or false ([ilmath]T[/ilmath] and [ilmath]F[/ilmath] respectively), then [ilmath]A[/ilmath] and [ilmath]B[/ilmath], denoted [ilmath]A\wedge B[/ilmath] has the following truth table:

[ilmath]\mathbf{A} [/ilmath]	[ilmath]\mathbf{B } [/ilmath]	[ilmath]\mathbf{A\wedge B } [/ilmath]
[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]F[/ilmath]
[ilmath]F[/ilmath]	[ilmath]T[/ilmath]	[ilmath]F[/ilmath]
[ilmath]T[/ilmath]	[ilmath]F[/ilmath]	[ilmath]F[/ilmath]
[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]

Negation

The only time [ilmath]\neg(A\wedge B)[/ilmath] is false is when [ilmath]A\wedge B[/ilmath] is true, which is only when [ilmath]A[/ilmath] it true and [ilmath]B[/ilmath] is true. All other cases [ilmath]\neg(A\wedge B)[/ilmath] is true, as [ilmath]A\wedge B[/ilmath] is false.

Thus we conclude: [ilmath]\neg(A\wedge B)\iff\big((\neg A)\vee(\neg B)\big)[/ilmath] where [ilmath]\vee[/ilmath] denotes logical or. This is true when either [ilmath]\neg A[/ilmath] or [ilmath]\neg B[/ilmath] is true. These are both false when both [ilmath]A[/ilmath] and [ilmath]B[/ilmath] are true. As the truth table will now show us.

Proof

By extending the table:

[ilmath]\mathbf{A} [/ilmath]	[ilmath]\mathbf{B } [/ilmath]	[ilmath]\mathbf{A\wedge B } [/ilmath]	[ilmath]\mathbf{\neg(A\wedge B)} [/ilmath]	Proof:	[ilmath]\mathbf{\neg A} [/ilmath]	[ilmath]\mathbf{\neg B} [/ilmath]	[ilmath]\mathbf{(\neg A)\vee(\neg B)} [/ilmath]	[ilmath]\mathbf{[\neg(A\wedge B)]\iff[(\neg A)\vee(\neg B)]} [/ilmath]
[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]T[/ilmath]		[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]
[ilmath]F[/ilmath]	[ilmath]T[/ilmath]	[ilmath]F[/ilmath]	[ilmath]T[/ilmath]		[ilmath]T[/ilmath]	[ilmath]F[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]
[ilmath]T[/ilmath]	[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]T[/ilmath]		[ilmath]F[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]
[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]T[/ilmath]	[ilmath]F[/ilmath]		[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]F[/ilmath]	[ilmath]T[/ilmath]