Majority function

Also called: majority, majority gate, majority vote, voting function, 2-of-3 majority

A function that outputs 1 when more than half of its inputs are 1. For three inputs it is AB + AC + BC.

The majority function outputs whatever most of its inputs say. With three inputs A, B and C it is 1 when at least two of them are 1:

Each product covers one pair being 1, and the OR collects them. Its truth table has 1s in rows 011, 101, 110 and 111.

It is a classic simplification exercise. The canonical form has four minterms, , and can combine with each of the other three. Copying it twice with the idempotent law gives the three two-literal products.

Where it shows up:

Voting and fault tolerance. Three copies of a circuit vote, and a single faulty copy is outvoted.
Adders. The carry out of a full adder is the majority of its three inputs: a carry happens when at least two of A, B and the carry in are 1.

Notice what the minimal form is not: ("exactly two") leaves out row 111 and has no simpler SOP.


0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	1
1	1	0	1
1	1	1	1

Worked examples

Example

Reading the expression row by row

Check on rows 101 and 100.

1.
Row 101 (A = 1, B = 0, C = 1): two inputs are 1, so the majority is 1. = 1, so the expression is 1. ✓
2.
Row 100 (A = 1, B = 0, C = 0): only one input is 1, so the majority is 0. = 0, = 0 and = 0, so the expression is 0. ✓
3.
Each product checks one pair. If any pair is all 1s, at least two inputs are 1.

Example

A carry is a majority

A full adder adds bits A, B and a carry in C. When is there a carry out?

1.
The column sum A + B + C (ordinary addition) ranges from 0 to 3.
2.
A carry out happens when that sum is 2 or 3, that is, when at least two of the bits are 1.
3.
So the carry out is the majority function, .
4.
Example: A = 0, B = 1, C = 1 sums to 2 = 10₂, carry 1. = 1. ✓

Common mistakes

Using each minterm only once and ending up with a term left over. must be reused.
Confusing "at least two" with "exactly two". The majority includes row 111.

Practice Majority function

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Learn it step by step

Majority function is taught in Boolean Algebra and Adders and ALUs.

Boolean Algebra →Adders and ALUs →

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