Question

: Show that the propositions (p ∨ q) ∧ (¬p ∨ r) and (p ∧ r) ⊕ (¬p ∧ q) are logically equivalent.

Answer 1

Answer:

$(p\vee q)\wedge(\neg p\vee r)\equiv(p\wedge r)\vee(\neg p\wedge q)$

Explanation:

The statements are logically equivalent if they have the same truth tables. So let´s use truth tables in order to determine if they are logically equivalent or not:

The picture that I attached you shows the truth table for each case. As you can see in the highlight columns:

$(p\vee q)\wedge(\neg p\vee r)\equiv(p\wedge r)\vee(\neg p\wedge q)$

They are  logically equivalent because they have exactly the same truth values between each other. Hence, we can conclude that they are logically equivalent.