Set Operations

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:

  1. A ∪ B = {x | x ∈ A ∨ x ∈ B}. The union of A and B.
  2. A ∩ B = {x | x∈ A ^ x ∈ B}. The intersection of A and B.
  3. 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:

Venn Diagram Union
A ∪ B is Shaded

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 AB, 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 ABC, 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:

Venn Diagram Intersection
A ∩ B is Shaded

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:

Venn Diagram set difference
A-B is shaded

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:
                                    Ac = {x ∈ U | x ∉ A}

EXAMPLE:

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

                                                A = {a, c, e, g}

                                    Then    Ac = {b, d, f} 

VENN DIAGRAM FOR COMPLEMENT:

Venn Diagram Complement
A Complement is Shaded

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.

Set Operations

Leave a comment

Your email address will not be published. Required fields are marked *