Question

Prove that (A⟹(B⟹C))⟺((A∧B)⟹C ) are tautologies:

(a) Using a true table (show columns for each individual component until you build the tautology to get credit)
(b) Using known tautology equivalence laws (e.g By De Morgan, Distribution, Associative, Commutative ...):

Answer 1

To prove the logical equivalence of (A⇒(B⇒C))⇔((A∧B)⇒C), one can either construct a truth table to show equivalence in all truth value conditions or apply tautology equivalence laws like commutative, distributive, and associative laws.

Step-by-step explanation:

The question involves proving a logical equivalence between two compound logical statements: (A⇒(B⇒C))⇔((A∧B)⇒C). This can be approached in two ways: using a truth table or by applying known tautology equivalence laws. For the purposes of this explanation, we'll focus on using a truth table.

• Firstly, we need to determine the truth values for each component of the propositions A, B, and C.
• Secondly, we will construct the truth values of the compound statements A⇒(B⇒C) and (A∧B)⇒C.
• Then, we compare these to show that they are indeed equivalent in all possible truth value situations, which would prove that the initial statement is a tautology.

The commutative property A+B=B+A is one of the logical equivalence rules that could be used to simplify the expressions, but for a complete proof, additional rules such as distributive or associative properties might be necessary.