Boolean Algebra Rules and Simplification

Overview

Recall questions on the notations and symbols for AND, OR, NOT, equivalence and exclusive OR, the commutative, associative and distributive laws, double negation, De Morgan's laws, basic identities, simplifying expressions and checking a simplification with a truth table.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   The same Boolean operation can be written in more than one way. What forms should you recognise?

   Model answer

   AND may appear as A.B, as AB with the dot omitted, as A AND B, or with a wedge symbol between the terms. OR may appear as A+B, as A OR B, or with a vee symbol. NOT may appear as a bar drawn over the term, as an apostrophe after it (A'), as a not-sign in front of it, or as NOT A. Exclusive OR is usually a plus inside a circle. Several notations are accepted, so you should be able to read any of them and you may answer in whichever you prefer, as long as you are consistent within an answer.

2. 2

   State the commutative law and give an example of it being used.

   Model answer

   Commutation says the order of the operands does not affect the result: A + B = B + A, and A.B = B.A. It is used to rearrange terms so that matching parts sit next to each other. For instance in B.A + A.B' you can commute the first term to A.B, giving A.B + A.B', which then factorises to A.(B + B') = A.1 = A - a simplification that was not visible before the rearrangement.

3. 3

   State the associative law and give an example of it being used.

   Model answer

   Association says that when the same operator is used throughout, the grouping does not affect the result: (A + B) + C = A + (B + C), and (A.B).C = A.(B.C). This means brackets can be moved or dropped in a chain of all-ANDs or all-ORs. For instance (A + B) + A' can be regrouped as (A + A') + B, and since A + A' = 1 and 1 + B = 1, the whole expression simplifies to 1. Note the law does not apply when AND and OR are mixed - then the brackets genuinely matter.

4. 4

   State the distributive law for AND over OR and give a worked example.

   Model answer

   A.(B + C) = A.B + A.C - the AND distributes across each term in the bracket, exactly as multiplication distributes over addition in ordinary algebra. Worked example: A.(A' + B) = A.A' + A.B. Since A.A' = 0 and 0 + anything is that thing, this reduces to A.B. Read in reverse the same law factorises, taking a common variable outside a bracket.

5. 5

   There is a second distributive law that has no counterpart in ordinary arithmetic. State it and show it works.

   Model answer

   A + (B.C) = (A + B).(A + C) - OR distributes over AND. It looks wrong because 2 + (3 x 4) is not (2+3) x (2+4), but in Boolean algebra it holds. Expanding the right-hand side gives A.A + A.C + B.A + B.C; since A.A = A this is A + A.C + A.B + B.C, and A absorbs both A.C and A.B, leaving A + B.C. It is worth memorising because it is the step that turns A + A'.B into (A + A').(A + B) = 1.(A + B) = A + B.

6. 6

   State the double negation rule and give an example of it being used.

   Model answer

   Negating a term twice returns it to its original value: NOT(NOT A) = A, written as A with two bars over it, or (A')' = A. It appears constantly after applying De Morgan's laws to an already-negated variable. For example NOT(A + B') becomes A'.(B')' by De Morgan, and the double negation on B then cancels to give A'.B.

Other ways to work this

The same content, four other ways.

What is inside Levo

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