Operations with sets

Operations with Sets

1. Union of Two Sets

The union of two or more sets is the set that contains all elements from the given sets, without repeating any element.

• The symbol for union is ∪.
• If set A and set B are given, then the union is written as: A ∪ B

Example 1

If A = {a, b} and B = {a, b, c, d, e},

Then:

A ∪ B = {a, b, c, d, e}

Example 2

If A = {a, b, c} and B = {5, 6, 2, 4},

Then:

A ∪ B = {a, b, c, 5, 6, 2, 4}

2. Complement of a Set

The complement of a set refers to all the elements in the universal set (U) that are not in the given set.

• If A is a subset of the universal set U, then the complement of A is written as: A′ or Ac

Example

Let the universal set U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

And let A = {1, 3, 5, 7, 9}

Then the complement of A is:

A′ = {0, 2, 4, 6, 8}

3. Intersection of Two Sets

The intersection of two sets is a new set that contains only the elements common to both sets.

• The symbol for intersection is ∩.
• If A and B are sets, then their intersection is written as: A ∩ B

Example 1

Let A = {a, b, d, e} and B = {a, b, d, f, g}

Common elements are: a, b, d

So:

A ∩ B = {a, b, d}

Example 2

Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}

There are no common elements, so:

A ∩ B = ∅ or { } (empty set)

4. Number of Elements in Union and Intersection

The number of elements in the union or intersection of two sets can be found by simply counting the elements in the resulting set after applying the operation.

Example

Let A = {a, b, c} and B = {5, 6, 2, 4}

• A ∪ B = {a, b, c, 5, 6, 2, 4} → Number of elements: 7
• A ∩ B = ∅ → Number of elements: 0