For statements p, q, and [tex]$r$[/tex], use a truth table to show that each of the following pairs of statements is logically equivalent: [tex]$P \Rightarrow(q \vee r) \text { and }(\neg r) \wedge P) \Rightarrow q \text {. }$[/tex]
Answer (2)
Construct a truth table with all possible combinations of truth values for P, q, and r.
Evaluate the truth values for q ∨ r and P ⇒ ( q ∨ r ) .
Evaluate the truth values for ¬ r , ( ¬ r ) ∧ P , and (( ¬ r ) ∧ P ) ⇒ q .
Compare the truth values of P ⇒ ( q ∨ r ) and (( ¬ r ) ∧ P ) ⇒ q to confirm they are identical, demonstrating logical equivalence. Logically Equivalent
Explanation
Understanding the Problem We are given two logical statements: P A rr ( q A rrr ) and (( ¬ r ) A rr P ) A rr q . We need to show that these two statements are logically equivalent using a truth table. Logical equivalence means that the two statements have the same truth value for all possible truth values of the variables P , q , and r .
Setting up the Truth Table We will construct a truth table with columns for P , q , r , q A rrr , P A rr ( q A rrr ) , ¬ r , ( ¬ r ) A rr P , and (( ¬ r ) A rr P ) A rr q . We will fill in all possible combinations of truth values for P , q , and r . There will be 2 3 = 8 rows.
Completing the Truth Table Here's the truth table:
P
q
r
q OR r
P -> (q OR r)
~r
(~r) AND P
((~r) AND P) -> q
T
T
T
T
T
F
F
T
T
T
F
T
T
T
T
T
T
F
T
T
T
F
F
T
T
F
F
F
F
T
T
F
F
T
T
T
T
F
F
T
F
T
F
T
T
T
F
T
F
F
T
T
T
F
F
T
F
F
F
F
T
T
F
T
Conclusion Comparing the columns for P ⇒ ( q ∨ r ) and (( ¬ r ) ∧ P ) ⇒ q , we see that the truth values are the same for all possible combinations of P , q , and r . Therefore, the two statements are logically equivalent.
Examples
Truth tables are fundamental in computer science for verifying the correctness of logical circuits and program logic. For instance, when designing a digital circuit, engineers use truth tables to ensure that the circuit's output matches the intended logical function for all possible input combinations. This ensures that the circuit behaves as expected under all conditions, preventing errors and ensuring reliable operation.
The truth table shows that the statements P ⇒ ( q ∨ r ) and (( ¬ r ) ∧ P ) ⇒ q have the same truth values for all combinations of truth values for P , q , and r . Thus, these two expressions are logically equivalent. By systematically creating the truth table, we demonstrated their logical equivalence clearly and accurately.
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