Prove the distributive law of Boolean algebra?

Answer: – The Distributive property is easy to remember, if you recall that “multiplication distributes over addition”. Formally, they write the property as a(b + c) = ab + ac”. In number, this means that 2(3+4) = 2×3+2×4. Any time they refer in a problem to using the distributive property, they want you to take something through the parentheses (or factor something out); any time a computer depends on multiplying through a parentheses (or factoring something out), they want you to  say that the computation used the distributive property.

  • Why is the following true? 2(x + y) = 2x + 2y

Since they distributed through the parentheses, this is true by the Distributive property.

  • Use the Distributive property to rearrange: 4x – 8

The Distributive property either takes something through a parentheses or else factors something out. Since there aren’t any parentheses to go into, you must need to factor out of. Then the answer is “By the Distributive property, 4x – 8 = 4(x – 2)”

“But wait!” you say. “The Distributive property says multiplication distributes over addition, not subtraction! What gives?” You make a good point. This is one of those times when it’s best to be flexible. You can either view the contents of the parentheses as the subtraction of a positive number (“x – 2”) or else as the addition of a negative number (“x + (-2)”). In the latter case, it’s easy to see that the distributive property applies, because you’re still adding; you’re just adding a negative.

The other two properties come in two versions each: one for addition and the other for multiplication. (Note that the Distributive property refers to both addition and multiplication, too, but both within just one rule.)