# Boolean absorption

A and ( A or C ) = A

And

A or A and C = A

How do these identities work?

Using the rule

A and ( B or C ) = A and B or A and C

For the first identity, I get

A and A or A and C = A or A and C

How is the C eliminated?

Mental substitution shows that it will equal A, but can it be algebraically shown.

Equation 2, too.

• Why Don't you just make a truth table? – joanolo Feb 27 '17 at 0:18
• Because, as I wrote in the question, I've already done it, mentally, and noticed that it works. I want to know why, and how it can be proven (without brute force). – Tobi Feb 27 '17 at 0:23
• What do you mean by "how to they work"? They don't "work"; they just are. What do you mean by "A or A and C"? Don't rely on everybody knowing what operator precedence you're using. – David Richerby Feb 27 '17 at 0:24
• AND, before, OR, David. – Tobi Feb 27 '17 at 0:26
• @Tobi, an extra pair of parentheses for explicit readability killed exactly no one. :) – kkm Mar 1 '17 at 6:56

Here is one way to prove the first identity: $$A \land (A \lor C) = (A \lor 0) \land (A \lor C) = A \lor (0 \land C) = A \lor 0 = A.$$ The second identity has a similar proof. Alternatively, you could use duality to deduce it from the first identity.