Question

1. not(not(a)) = a

incorrect
correct
the not of not(a) is just a. ex: when a is true, (not(not(a))) = not(not(true)) = not(false) = true, which was as original value. the same happens when a is false.

1. not(not(a)) and b

= not(not(a)) or not(b)
= a or not(b)
incorrect
correct

Explanation:

Step1: Recall double - negation law

In logic, the double - negation of a statement is equivalent to the original statement. If we have a logical variable \(A\), then \(\text{NOT}(\text{NOT}(A))=A\). This is because if \(A\) is true, \(\text{NOT}(A)\) is false, and \(\text{NOT}(\text{NOT}(A))\) is true (the opposite of false). Similarly, if \(A\) is false, \(\text{NOT}(A)\) is true, and \(\text{NOT}(\text{NOT}(A))\) is false (the opposite of true).

Step2: Analyze each option

• For option "NOT(NOT(A))", by the double - negation law, it is equal to \(A\).
• For "NOT(NOT(A)) AND B", this is a conjunction. It will be true only when both \(\text{NOT}(\text{NOT}(A))\) (which is \(A\)) and \(B\) are true. It is not always equal to \(A\).
• For "NOT(NOT(A)) OR NOT(B)", this is a disjunction. Its truth - value depends on the values of \(A\) and \(B\) and is not always equal to \(A\).
• For "A OR NOT(B)", its truth - value depends on the values of \(A\) and \(B\) and is not always equal to \(A\).

Answer:

The correct option is NOT(NOT(A)) as it is equal to \(A\) according to the double - negation law in logic.