**A. Notes**

In this last lesson of boolean algebra, we learned some common **identities**. Identities refer to the things that are always the same. For instance, A+B and B+A are always the same, so A+B and B+A are identities, we write A+B=B+A.

Some commonly used identities in Boolean Algebra are below.

**#9** above is also called **Law of Tautology**. We can use these identities to **simplify** compound boolean expressions. While applying these identities, please keep in mind:

A

*****or a**dot**both means**AND**, and sometimes we even**omit them**. i.e. AB, A*B, A&&B all mean A AND B.A

**bar**on top of a variable means**NOT**, just as**!**and**~**. A bar on top of a compound expression means to negate the whole expression.

See the example below. **Solution I** and **II** both simplify this boolean expression.

**B. HW**

Simplify the following boolean expressions:

2. Use DeMorgan's Law to simplify the following:

true

A + B

not(A)+C+B

not(A) x B

not(A) x B

C x B + C x A

A x B + A x C

B

True

not (A) x B x C

(A or B) or (not B)

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

(not A) or C or (A and B)

(not A) and (B or (A and C))

(not A) and (B or A)

(B or (A and (not B)) and (C or (A and C))

(A and B and C) or (A and (not B) and C) or (A and B and (not C))

(B or (A and C)) and (B or (not A))

~A + ~B + ~C

A + ~B + ~C