For statements [tex]$p, q$[/tex], and [tex]$r$[/tex], show that the following compound statements are tautologies.
a. [tex]$p \Rightarrow(p \vee q)$[/tex].
b. [tex]$(p \wedge(p \Rightarrow q)) \Rightarrow q$[/tex].
c. [tex]$((p \Rightarrow q) \wedge(q \Rightarrow r)) \Rightarrow(p \Rightarrow r)$[/tex].

Construct truth tables for each compound statement.
Verify that the final column in each truth table (representing the compound statement) is always True.
Conclude that each statement is a tautology because it is always True.
Therefore, p ⇒ ( p ∨ q ) , ( p ∧ ( p ⇒ q )) ⇒ q , and (( p ⇒ q ) ∧ ( q ⇒ r )) ⇒ ( p ⇒ r ) are tautologies. T r u e ​

Explanation

Understanding Tautologies We need to show that the given compound statements are tautologies. A tautology is a compound statement that is always true, regardless of the truth values of the individual statements. We will construct truth tables for each statement to verify this.

Analyzing Statement a Let's analyze the first statement: p ⇒ ( p ∨ q ) . We need to create a truth table with columns for p , q , p ∨ q , and p ⇒ ( p ∨ q ) .

Truth Table for Statement a Here's the truth table for p ⇒ ( p ∨ q ) :





p
q
p \vee q
p \Rightarrow (p \vee q)



True
True
True
True


True
False
True
True


False
True
True
True


False
False
False
True


Since the last column is always True, the statement p ⇒ ( p ∨ q ) is a tautology.

Analyzing Statement b Now let's analyze the second statement: ( p ∧ ( p ⇒ q )) ⇒ q . We need to create a truth table with columns for p , q , p ⇒ q , p ∧ ( p ⇒ q ) , and ( p ∧ ( p ⇒ q )) ⇒ q .

Truth Table for Statement b Here's the truth table for ( p ∧ ( p ⇒ q )) ⇒ q :





p
q
p \Rightarrow q
p \wedge (p \Rightarrow q)
(p \wedge (p \Rightarrow q)) \Rightarrow q



True
True
True
True
True


True
False
False
False
True


False
True
True
False
True


False
False
True
False
True


Since the last column is always True, the statement ( p ∧ ( p ⇒ q )) ⇒ q is a tautology.

Analyzing Statement c Finally, let's analyze the third statement: (( p ⇒ q ) ∧ ( q ⇒ r )) ⇒ ( p ⇒ r ) . We need to create a truth table with columns for p , q , r , p ⇒ q , q ⇒ r , ( p ⇒ q ) ∧ ( q ⇒ r ) , p ⇒ r , and (( p ⇒ q ) ∧ ( q ⇒ r )) ⇒ ( p ⇒ r ) .

Truth Table for Statement c Here's the truth table for (( p ⇒ q ) ∧ ( q ⇒ r )) ⇒ ( p ⇒ r ) :





p
q
r
p \Rightarrow q
q \Rightarrow r
(p \Rightarrow q) \wedge (q \Rightarrow r)
p \Rightarrow r
((p \Rightarrow q) \wedge (q \Rightarrow r)) \Rightarrow (p \Rightarrow r)



True
True
True
True
True
True
True
True


True
True
False
True
False
False
False
True


True
False
True
False
True
False
True
True


True
False
False
False
True
False
False
True


False
True
True
True
True
True
True
True


False
True
False
True
False
False
True
True


False
False
True
True
True
True
True
True


False
False
False
True
True
True
True
True


Since the last column is always True, the statement (( p ⇒ q ) ∧ ( q ⇒ r )) ⇒ ( p ⇒ r ) is a tautology.

Conclusion All three compound statements are tautologies because their truth tables show that they are always true, regardless of the truth values of their individual components.

Examples
In computer science, tautologies are used to verify the correctness of logical circuits and programs. For example, if a program's logic can be expressed as a tautology, it guarantees that the program will always produce a correct result, regardless of the input. This is particularly useful in critical systems where errors can have severe consequences, such as in medical devices or aerospace systems. By ensuring that the underlying logic is a tautology, engineers can build more reliable and robust systems.

Answered by GinnyAnswer | 2025-07-04