For a Boolean variable 𝑥, which of the following statements is/are FALSE?

Solution:

Evaluate each statement for a Boolean variable x:

• x · 1 = x: True. 1 is the identity for AND, so true AND x equals x.

• x + 1 = x: False. 1 represents logical TRUE, and TRUE OR anything is TRUE, so x + 1 = 1, not x.

• x · x = 0: False. By the idempotent law, x AND x equals x, so x · x = x, not 0.

• x + not x = 1: True. A variable OR its complement is always TRUE (the complement law).

Therefore, the statements that are FALSE are: x + 1 = x and x · x = 0.