Logical and

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

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Definition

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

$\mathbf{A}$	$\mathbf{B }$	$\mathbf{A\wedge B }$
$F$	$F$	$F$
$F$	$T$	$F$
$T$	$F$	$F$
$T$	$T$	$T$

Negation

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

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

Proof

By extending the table:

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