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Expert-verified Found in: Page 35 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # Show that each of these conditional statements is a tautology by using truth tables.(a) ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ (b) ${\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\vee }}{\mathbf{q}}{\mathbf{\right)}}$ (c) role="math" localid="1668154499853" ${\mathbf{¬}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ (d)${\mathbf{\left(}}{\mathbit{p}}{\mathbf{\wedge }}{\mathbit{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbit{p}}{\mathbf{\to }}{\mathbit{q}}{\mathbf{\right)}}$(e) ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ (f)${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$

It is shown that the conditional statement ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$is a tautology as the output of the truth table consists of only T.

See the step by step solution

## Definition of Truth Tables

A logic gate truth table depicts each feasible input sequence to the gate or circuit, as well as the resulting output based on the combination of these inputs.

## To Show conditional statement is a tautology using a truth table

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

Prepare the truth table for ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}{\mathbf{.}}$

 p q localid="1668156225147" ${\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}$ ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ T T T T T F F T F T F T F F F T

Truth Table

The conditional statement ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ is a tautology as the output of the truth table consists of only T.

(b) ${\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\vee }}{\mathbf{q}}{\mathbf{\right)}}$

Prepare the truth table for ${\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\vee }}{\mathbf{q}}{\mathbf{\right)}}$ .

 p q ${\mathbf{p}}{\mathbf{\vee }}{\mathbf{q}}$ ${\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\vee }}{\mathbf{q}}{\mathbf{\right)}}$ T T T T T F T T F T T T F F F T

Truth Table

The conditional statement is a tautology as the output of the truth table consists of only T.

(c) ${\mathbf{¬}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$

Prepare the truth table for ${\mathbf{¬}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$

 p q ${\mathbf{¬}}{\mathbf{p}}$ ${\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}$ ${\mathbf{¬}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ T T F T T T F F F T F T T T T F F T T T
Truth Table

The conditional statement ${\mathbf{¬}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ is a tautology as the output of the truth table consists of only T.

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

Prepare the truth table for localid="1668162884798" ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{.}}$

 p q ${\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}$ ${\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}$ localid="1668163037369" ${\mathbf{\left(}}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ T T T T T T F F F T F T F T T F F F T T

Truth Table

The conditional statement ${\left(}{\mathbf{p}}{\mathbf{\wedge }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ is a tautology as the output of the truth table consists of only T.

(e) ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$

Prepare the truth table for ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ .

 p q ${\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}$ ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{p}}$ T T T F T T F F T T F T T F T F F T F T

Truth Table

The conditional statement ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$ is a tautology as the output of the truth table consists of only T.

(f) ${¬}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$

Prepare the truth table for ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$.

 p q ${\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}$ ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}$ ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$ T T T F T T F F T T F T T F T F F T F T

Truth Table

The conditional statement ${\mathbf{¬}}{\mathbf{\left(}}{\mathbf{p}}{\mathbf{\to }}{\mathbf{q}}{\mathbf{\right)}}{\mathbf{\to }}{\mathbf{¬}}{\mathbf{q}}$is a tautology as the output of the truth table consists of only T.

Therefore, it has been shown that the given conditional statements are tautologies using truth tables. ### Want to see more solutions like these? 