Which of the following represents the Idempotent law in Boolean Algebra?
2018
Which of the following represents the Idempotent law in Boolean Algebra?
Answer: B. A + A = A — Concept — In Boolean algebra an operation is called idempotent when it is applied to two copies of the same operand and returns that operand unchanged. For…
- A.
A + 0 = A
- B.
A + A = A
- C.
A + A' = 1
- D.
(A')' = A
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Correct answer: B
Concept — In Boolean algebra an operation is called idempotent when it is applied to two copies of the same operand and returns that operand unchanged. For every Boolean variable x this means x + x = x under OR and x · x = x under AND. Repeating one variable on both sides of the operator therefore adds no new information to the result.
Applying it here — Test the pattern “the same variable on both sides of the operator” against each Boolean value of A:
A | A + A | A · A |
|---|---|---|
0 | 0 | 0 |
1 | 1 | 1 |
Both result columns reproduce the value in the A column, so A + A = A holds for every Boolean value of A, and its AND form A · A = A is the same law.
Contrast with the other expressions —
A + 0 = A: the second operand is the constant 0 rather than a repeat of the variable, which is the identity law for OR.
A + A' = 1: the second operand is the complement of the first, so the result is the constant 1, which is the complement law.
(A')' = A: a single variable is complemented twice and no binary operator is involved at all, which is the involution or double-negation law.
The expression that states the idempotent law is therefore A + A = A.