Venn Diagram


Venn diagram  is a diagram that shows all possible logical relations between a finite collection of different sets

Introduction

Venn charts, named after the English mathematician John Venn, which introduced their use in 1881, Venn diagram can be used to represent sets graphically. In Venn diagrams the universal set U which contains all the subjects that are shown as a rectangle.

In this rectangle, the circles or other geometric figures are used to represent the sets.( Note that the universal sets vary depending on which objects are of interest.) Sometimes points represent the specific elements of the set

These diagrams shows elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve represent elements of the set S, while points outside the boundary represent elements not in the set S.[wikipedia]

Venn diagrams can be used to illustrate that a set A is a subset of a set B. We draw the universal set S as a rectangle.Within this rectangle we draw a circle for B and a circle for A .

Venn diagram
Venn diagram

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

EXAMPLES:

(1) If A = {1, 3, 5, 7} and B = {1, 2, 4, 6} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

(2)

A = {x is an even integer larger than 1}

B = {x is an odd integer larger than 1}

[latex]{\displaystyle A\cup B=\{2,3,4,5,6,\dots \}}[/latex]

(3)                                    Let U = {1, 2, 3, 4, 5, 6, 7}

                                 A = {1, 3, 5, 7},           B = {2, 4, 5, 6}

                                Then     A ∪ B = {x ∈U | x ∈A or x ∈ B}

                                                      ={1, 3, 4, 5 ,6, 7}

VENN DIAGRAM FOR UNION:

Venn Diagram Union
A ∪ B is Shaded

MEMBERSHIP TABLE FOR UNION:

Membership table 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 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]

VENN DIAGRAM FOR INTERSECTION:

Venn Diagram Intersection
A ∩ B is Shaded

MEMBERSHIP TABLE FOR INTERSECTION:

Membership table for intesection

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

MEMBERSHIP TABLE FOR SET DIFFERENCE:

Membership table 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:
                                    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

MEMBERSHIP TABLE FOR COMPLEMENT:

Membership table for complement

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