Logical and

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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]


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[Note 1] [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.


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]


  1. English "and"