# Why is Ā+AB = Ā+B ?

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104 # Why is Ā+AB = Ā+B ?

Ā+AB = Ā+B because the terms AB and B are equal. This follows from the fact that A+A = A, which is one of the axioms of Boolean algebra.

In Boolean algebra, A+A = A is known as the “absorption” or “idempotent” law. It states that the OR of a value with itself is equal to the value itself. Similarly, A*A = A is known as the “idempotent” law of AND.

Here’s a proof that AB = B using the axioms of Boolean algebra:

1. A+A = A (absorption law)
2. AB+A = A (substitution, using step 1)
3. AB+B = B (absorption law)
4. AB = B (cancellation law)

Here are some examples to illustrate the absorption law in Boolean algebra:

A+A = A

For example, if A represents the statement “It is raining outside,” then A+A would represent the statement “It is raining outside, OR it is raining outside.” This is clearly true, so A+A = A.

AB+A = A

For example, if A represents the statement “It is raining outside,” and B represents the statement “I have an umbrella,” then AB+A would represent the statement “It is raining outside AND I have an umbrella, OR it is raining outside.” This is also true, because if it is raining outside and I have an umbrella, then it is necessarily true that it is raining outside. Therefore, AB+A = A.

AB+B = B

For example, if A represents the statement “It is raining outside,” and B represents the statement “I have an umbrella,” then AB+B would represent the statement “It is raining outside AND I have an umbrella, OR I have an umbrella.” This is true because if it is raining outside and I have an umbrella, then I have an umbrella (which is one of the terms being ORed). Similarly, if it is not raining outside, then having an umbrella is still true. Therefore, AB+B = B.