A logical binary relation $, is defined as follows: A$B is True when (A,B) = (True,True); True when (A,B) = (True,False)...

Question

A logical binary relation $, is defined as follows: A$B is True when (A,B) = (True,True); True when (A,B) = (True,False); False when (A,B) = (False,True); True when (A,B) = (False,False). Let ~ be the unary negation (NOT) operator, with higher precedence than $. Which one of the following is equivalent to A AND B? A. (~A $ B)  B. ~(A $ ~B)  C. ~(~A $ ~B)  D. ~(~A $ B)

Accepted Answer

~(~A $ B) = A AND B  (Option D) Read the truth table for $: A$B is False only in the single row (A=False, B=True) and True everywhere else. That pattern is exactly 'A OR NOT B', equivalently 'B implies A'. To match A AND B (which is True only when both are True) you negate the right output once and feed in ~A: ~A $ B works out to ~A OR ~B = NOT(A AND B), so negating that whole thing gives back A AND B.