# Difference of Sets

The difference of set *B* from set *A, *denoted by *A-B,* is the set of all the elements of set *A* that are not in set *B*. In mathematical term,

A-B = { x: x∈A and x∉B}

If *(A∩B)* is the intersection between two sets *A* and *B* then,

A-B = A - (A∩B)

## Difference of Sets Example

### Example #1

**If A = { a, b, c, d, e } and B = { a, e, f, g}, find A-B and B-A.**

The elements in **only A** are **b, c, d** and elements in **only B** are **f, g**. Thus,

A-B = {b, c, d} and B-A = {f, g}

Notice that, *A-B* may not be equal to *B-A.*

### Example #2

**If A = { 1, 2, 4, 6, 8} and A-B = {1, 6, 8}, Find A∩B.**

The intersection of A and B, (A∩B) is the set of all elements common in both A and B. Thus,

A∩B = A - (A-B) or, A∩B = {2, 4}

### Identities Involving Difference of Sets

- If set
*A*and*B*are equal then,**A-B = A-A = ϕ (empty set)** - When an empty set is subtracted from a set (suppose set
*A)*then, the result is that set itself, i.e,**A - ϕ = A**. - When a set is subtracted from an empty set then, the result is an empty set, i.e,
**ϕ - A = ϕ**. - When a superset is subtracted from a subset, then result is an empty set, i.e,
**A - B = ϕ if A ⊂ B**

- If A and B are disjoint sets then,
**A-B = A**and**B-A = B**