Which of the following boolean algebra statements is true?

a) x & y = ~x | ~y

b) x | y | z = x & (y | z)

c) x | (y & z) = (x & y) | (x & z)

d) x & (y | z) = (x & y) | (x & z)

The only statement that is True is the d) x & (y | z) = (x & y) | (x & z)

Explanation:

The statement x & (y | z) = (x & y) | (x & z) satisfies the Distributive Law of Boolean Algebra that states:

A & (B | C) = (A & B) | (A & C).

You can also check that the others are False if you use the operation notation, lets see that any of the other statements are true:

Statement a) x & y = ~x | ~y is equivalent to x * y=~x + ~y and this is False

Statement b) x | y | z = x & (y | z) is equivalent to x + y + z = x * (y + z), by solving this you have x + y + z = x * y + x * z and those expressions are not the same, so is False

For Statement c) x | (y & z) = (x & y) | (x & z) we have that this one does not satisfies the Distributive Law, therefore is False