Boolean algebra study guide

What simplification means

A Boolean simplification is a sequence of equivalent expressions. Each step must preserve the output for every possible combination of Boolean input values. The aim is not merely to produce a shorter-looking expression: the final expression must have the same logical behaviour as the original one.

Read an expression by identifying its outermost operation and its parentheses. For example, in ¬(A + B), the NOT applies to the complete OR expression. In ¬A + B, the NOT applies only to A. These are structurally different expressions, so they must not be transformed in the same way.

A method for solving expressions

1. Mark the structure. Identify grouped terms, NOT operations, AND operations and OR operations.
2. Look for direct identities. Search for 0, 1, repeated terms, complementary terms and double negation.
3. Apply De Morgan’s law when a NOT is applied to a group. Change the operator and negate each term.
4. Use distributive or reverse-distributive forms to expose a simpler identity. Factoring can reveal A + ¬A = 1 or A · ¬A = 0.
5. Continue until no justified simplification is available. Keep the equality sign between each stage.

Worked reasoning with De Morgan’s law

Consider ¬(A · ¬B). The NOT applies to an AND group, so use ¬(X · Y) = ¬X + ¬Y, with X = A and Y = ¬B:

¬(A · ¬B) = ¬A + ¬(¬B)

Now apply double negation:

¬A + ¬(¬B) = ¬A + B

Thus the expression is simplified to ¬A + B. The second term becomes B because the two NOT operations cancel.

Worked reasoning with complementary terms

Consider (A + B) · (A + ¬B). Apply the distributive identity:

(A + B) · (A + ¬B) = A + (B · ¬B)

The product B · ¬B is 0, because a Boolean value cannot be both true and false in that product. Therefore:

A + (B · ¬B) = A + 0 = A

This example shows why selecting the correct direction of distribution matters: expanding or factoring can expose a complement identity that was not initially visible.

Distinctions to retain

The complement identities are different from idempotent identities. A + A = A, but A + ¬A = 1; similarly, A · A = A, but A · ¬A = 0. De Morgan’s laws are also not the same as simply distributing NOT: the operator changes from AND to OR, or from OR to AND.

Self-check guidance

After simplifying, inspect every equality step. Ask: what is the outermost operation, which identity has been used, and have all parentheses and NOT symbols been retained? For an additional check, choose Boolean values for the variables and evaluate both the original and final forms. Matching results for selected values can support your reasoning, but the algebraic identity used in every step is the formal justification. Practise both directions of distributive identities and expressions in which De Morgan’s laws must be followed by double negation.