# Negation of implies

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Here is the truth table for implies, with the negation computed (the negation is simply [ilmath]\neg(A\implies B)[/ilmath] - so [ilmath]\neg[/ilmath] of whatever is in the [ilmath]A\implies B[/ilmath] column)

[ilmath]A[/ilmath] | [ilmath]B[/ilmath] | [ilmath]A\implies B[/ilmath] | [ilmath]A[/ilmath] | [ilmath]\neg B[/ilmath] | [ilmath]A\wedge(\neg B)[/ilmath] | [ilmath]\neg(A\implies B)[/ilmath] | |
---|---|---|---|---|---|---|---|

T | T | ||||||

T | T | ||||||

T | T | T | T | T | |||

T | T | T | T |

It is easy to see that the negation of implies is the same as [ilmath]A\wedge(\neg B)[/ilmath]

So to negate [ilmath]A\implies B[/ilmath] we require to have [ilmath]A[/ilmath] but not [ilmath]B[/ilmath]