Simplifying Boolean Algebra

Numbers and formulas help us put randomness into order. Electronics is no exception, and Boolean algebra takes confusing circuit diagrams and explains their functions mathematically. Boolean algebra serves as a way to quickly design circuits without drawing them out by using simple algebraic formulas.

Difficulty:

Moderate

Instructions

1. Start with this simple equation. A + 0 = A; as with regular math, adding zero to a number will equal the original number. When thinking in Boolean terms, see A + 1 as being equal to 1. Know that "A" could be any number, then adding it to its complimentary number, A', still equals "1." Write the formula as (A+A') = 1. Keeping your mind on Boolean terms; see "A" added to "A" as being equal to A, or A + A = A (see Tips below).

2. Know the following Boolean multiplicative identities by heart: (0)(A)=0, (1)(A)=A, (A)(A)=A, and (A)(A')=0. Memorize these Boolean additive identities: (A + 0)=A, (A+1)=1, (A + A) = A, (A + A')=1. Understand that when you invert a variable two times, you get the same variable (see Tips Below).

3. See (A + B) as being equal to (B + A). Visualize a circuit receiving an "A" and a "B" feed; it doesn't matter how these feeds are arranged, the result is the same. Know that Boolean algebra has commutative property of multiplication rules like regular Algebra. See (A)(B) as being equal to (B)(A). Know these two formulas well: A + (B + C) = (A + B) + C; "A" times (B)(C)= (B)(C) times "C" (see Tips below).

4. Memorize the formulas in the first three steps, then simplify Boolean algebra expressions. For example, factor "A" out of this Boolean expression: A + AB = A to get this formula: A(1+B)=A. Replace "B" in the equation with "A" in this example, as A could be any variable; you get A(1+A)=A. Replace (1+A) with "1" since (1+A)=1; you get 1A. Substitute "A" for 1A since 1A=A; you get "A" in place of A + AB (see Resources below).

5. Use the Boolean algebra distributive rule to simplify (A+B)(A+C). Expand for your first step to get AA+AC+AB+BC as you would in regular algebra. Covert the first variable, AA, to "A" based on the formula A=AA. Regroup A+AC+AB+BC to read A+AB+AC+BC. Utilize the formula A=A+AB to get A+AC+BC. Know that A+AC is the same thing as A+AB, which is the same thing as "A." Replace A+AC+BC with A+BC (see Resources below).

Tips & Warnings

• Understand the basic reason Boolean identities take their form; some of them are consistent with basic algebra, others are consistent with how circuits work. Like regular algebra, Boolean algebra has associative, distributive and commutative properties. Think in circuit terms when looking at A + 1 = 1. One represents a signal that superimposes whatever number "A" happens to be. When you're connecting "A" to another "A," you're still getting "A." Think of connecting two party string lights. Imagine both string lights having hot peppers; after connecting both string lights, you still have a string light of hot peppers. To understand double inversion, invert the number 2 to get ½. Invert ½ and you'll end up with 2 again. But, unlike algebra, multiplying these two numbers results in zero, not one. This is due to a canceling effect. This article uses a single apostrophe to represent a bar written over the top of a variable. Draw the bars over the variables if you're working Boolean functions on paper.

• Many Boolean algebra principles are similar to algebra, which means they could be simplified similar to regular algebra. But Boolean algebra is also different. As seen in the above steps, "A" could be any number compatible to "B." This makes it possible for like formulas, A+AB, A+AC, to equal the same variable, A. These are based on circuit behaviors; instead of on mathematical principles.

Resources

• Basic Boolean formulas to memorize
• Boolean algebra's commutative properties
• Rules of simplification