Question

You are given the following logical expression: (P ∧ Q) ∨ (~P ∧ R). Prove that the left-hand side is equivalent to the right-hand side by logical equivalence. Submit your work and be sure to label which identity or law you used in each step.

a. Comμtative Law
b. Distributive Law
c. Identity Law
d. De Morgan's Law

Answer 1

To prove that the left-hand side (LHS) is equivalent to the right-hand side (RHS) of the logical expression (P ∧ Q) ∨ (~P ∧ R), we can use the distributive, identity, and commutative laws. By applying these laws step-by-step, we can simplify the LHS to match the RHS.

Step-by-step explanation:

To prove that the left-hand side (LHS) is equivalent to the right-hand side (RHS) of the logical expression (P ∧ Q) ∨ (~P ∧ R), we can use the logical equivalence laws. Let's break it down step-by-step:

1. Distributive Law: Start by applying the distributive law to the conjunction (P ∧ Q) with the disjunction (~P ∧ R):

(P ∧ Q) ∨ (~P ∧ R)

= (P ∨ ~P) ∧ (P ∨ R) ∧ (Q ∨ ~P) ∧ (Q ∨ R)

1. Identity Law: Use the identity law to simplify:

(P ∨ ~P) ∧ (P ∨ R) ∧ (Q ∨ ~P) ∧ (Q ∨ R)

= true ∧ (P ∨ R) ∧ (Q ∨ ~P) ∧ (Q ∨ R)

1. Commutative Law: Apply the commutative law to rearrange the conjunctions:

true ∧ (P ∨ R) ∧ (Q ∨ ~P) ∧ (Q ∨ R)

= (P ∨ R) ∧ true ∧ (Q ∨ ~P) ∧ (Q ∨ R)

1. Identity Law: Simplify further:

(P ∨ R) ∧ true ∧ (Q ∨ ~P) ∧ (Q ∨ R)

= (P ∨ R) ∧ (Q ∨ ~P) ∧ true ∧ (Q ∨ R)

1. Commutative Law: Rearrange the conjunctions again:

(P ∨ R) ∧ (Q ∨ ~P) ∧ true ∧ (Q ∨ R)

= (P ∨ R) ∧ (Q ∨ R) ∧ true ∧ (Q ∨ ~P)

1. Identity Law: Simplify one more time:

(P ∨ R) ∧ (Q ∨ R) ∧ true ∧ (Q ∨ ~P)

= (P ∨ R) ∧ (Q ∨ R) ∧ (Q ∨ ~P)

Therefore, we have shown that the LHS (P ∧ Q) ∨ (~P ∧ R) is equivalent to the RHS (P ∨ R) ∧ (Q ∨ R) ∧ (Q ∨ ~P) by applying the distributive, identity, and commutative laws.