Set operations can be performed in many different ways, they can be performed by combining two or more sets.

**For example**, starting with a set of majors of mathematics in your school and a set of informatics majors at your college, we can train students who are majors in mathematics or informatics, a set of students who are mathematics and IT collaborators, and a set of students who do not study mathematics.

If we think of sets as representing , each logical connective that gives

rise to a corresponding set operations:

- A ∪ B = {x | x ∈ A ∨ x ∈ B}. The
**union**of A and B. - A ∩ B = {x | x∈ A ^ x ∈ B}. The
**intersection**of A and B. - A \ B = {x | x ∈ A ^ x ∉ B}. The set
**difference**of A and B.

(Of these, union and intersection are the most important in practice.)

Corresponding to implication is the notion of a subset:

**A ⊆ B **(“A is a subset of B”) if and only if

**8x : x 2→A ! x ∈ B. **

## Union

Let A and B be subsets of a universal set U. The union of sets A and B is the set of all elements in U that belong to A or to B or to both, and is denoted A ∪ B.

Symbolically:

A ∪ B = {x ∈U | x ∈A or x ∈ B

**EXAMPLE:**

The union of the sets **{1, 3, 5}** and **{1, 2, 3}** is the set **{1, 2, 3, 5}**

that is,

**{1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}.**

**VENN DIAGRAM FOR UNION:**

## INTERSECTION:

Let A and B subsets of a universal set U. The intersection of sets A and B is the set of all elements in U that belong to both A and B and is denoted as A ∩ B.Symbolically:

A ∩ B = {x ∈U | x ∈ A and x ∈B}

Intersection is an associative operation; that is, for any sets *A*, *B*, and *C*, one has *A* ∩ (*B* ∩ *C*) = (*A* ∩ *B*) ∩ *C*. Intersection is also commutative; for any *A* and *B*, one has *A* ∩ *B* = *B* ∩ *A.* It thus makes sense to talk about intersections of multiple sets. The intersection of *A*, *B*, *C*, and *D*, for example, is unambiguously written *A* ∩ *B* ∩ *C* ∩ *D*. [Wikipedia]

**EXAMPLE:**

The intersection of the sets **{1, 3, 5}** and **{1, 2, 3}** is the set **{1, 3}**

that is,** {1, 3, 5} ∩ {1, 2, 3} = {1, 3}.**

**VENN DIAGRAM FOR INTERSECTION:**

## Disjoint

Two sets are called disjoint if their intersection is the empty set.

**EXAMPLE:**

Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}.

Because A ∩ B = ∅, A and B are disjoint

**DIFFERENCE**

Let A and B be subsets of a universal set U. The difference of “A and B” (or relative complement of B in A) is the set of all elements in U that belong to A but not to B, and is denoted A – B or A \ B.

Symbolically:

A – B = {x ∈U | x ∈ A and x ∈B}

**EXAMPLE:**

Let U = {a, b, c, d, e, f, g}

A = {a, c, e, g}, B = {d, e, f, g}

Then A – B = {a, c}

**VENN DIAGRAM FOR SET DIFFERENCE:**

## COMPLEMENT:

Let A be a subset of universal set U. The complement of A is the set of all element in U that do not belong to A, and is denoted AN, A or Ac

Symbolically:

A^{c}= {x ∈ U | x ∉ A}

**EXAMPLE:**

Let U = {a, b, c, d, e, f, g]

A = {a, c, e, g}

Then A^{c} = {b, d, f}

**VENN DIAGRAM FOR COMPLEMENT:**

## Set Identities:

Following Lists **key identities** of the set. There are three different methods to prove several of those **identities**. These methods are presented to show that many **approaches** to solving a problem often exist. Set identities and propositional equivalence are only special cases of identity for **Boolean algebra** as evidence of the remaining identities.