Question

Determine whether$\mathbf{(}\mathbf{\neg }\mathbf{q}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{\left)}\mathbf{\right)}\mathbf{\to }\mathbf{-}\mathbf{p}$is a tautology

Short answer

The given conditional statement$\mathbf{(}\mathbf{\neg }\mathbf{q}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{\left)}\mathbf{\right)}\mathbf{\to }\mathbf{-}\mathbf{p}$ is solved and this given statement is a tautology.

Step-by-step solution

Definition

A proposition is a tautology, if the proposition is always true (thus true for any combination of truth values for the variablesp, q, r,...)

TRUTH TABLE

A conjunction $\mathbf{p}\mathbf{\wedge }\mathbf{q}$is true, if both (sub) propositions (pandq) are true.

A disjunction $\mathbf{p}\mathbf{\vee }\mathbf{q}$is true, if either of the (sub) propositions (porq) are true.

A negation $\mathbf{\neg }\mathbf{p}$is true, if the (sub) proposition is false.

A conditional statement $\mathbf{p}\mathbf{\to }\mathbf{q}$is true, if pis false, or if both (sub) propositions are true.

A bi-conditional statement $\mathbf{p}\mathbf{\leftrightarrow }\mathbf{q}$is true, if both (sub) propositions are true or if both (sub) propositions are false.

Solution

$\mathbf{(}\mathbf{\neg }\mathbf{q}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{\left)}\mathbf{\right)}\mathbf{\to }\mathbf{-}\mathbf{p}$

The conditional statement is a tautology; because the last column in the following truth table contains only trueTvalues

p	q	$\mathbf{\neg }\mathbf{p}$	$\mathbf{\neg }\mathbf{q}$	$\mathbf{p}\mathbf{\to }\mathbf{q}$	$\mathbf{-}\mathbf{p}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}$	$\mathbf{(}\mathbf{\neg }\mathbf{q}\mathbf{\wedge }\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{\left)}\mathbf{\right)}\mathbf{\to }\mathbf{p}$
T	T	F	F	T	F	T
T	F	F	T	F	F	T
F	T	T	F	T	F	F
F	F	T	T	F	T	T