Set Theory Venn Diagrams

INTRODUCTION TO SET THEORY VENN DIAGRAMS:

Define Set:

A set is a collection of well defined objects and these things which constitute a set are called its ‘elements’ or ‘members’.

Example:Set of odd numbers from 1 to 10 = {1, 3, 5, 7, 9}

Define Set theory - Venn diagram:

The geometrical representation of different types of sets is called “Venn diagram”

Example:

Basic Operations on Sets:

The primary operation on a set is,

- Union of sets:

The union of two nonempty sets A and B is the set consisting of all elements belonging to A or B or to both of them.

Ex: A = {1, 2, 5, 6} and B= {3, 7, 8, 9}

A `uu` B= {1, 2, 3, 5, 6, 7, 8, 9}

- Intersection of sets:

Let P and Q are two sets. The set of elements present in both the sets A and B are known as the set intersection of P and Q.

Ex P = {1, 2, 4, 6, 8}, Q = {6, 8, 9, 10, 12}.

P  `nn` Q = {6, 8}.

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Set Theorey Notations Used in Venn Diagrams

Let A, B are two sets and the following are the notations that are used in Venn diagrams:

AUB – >A union B which means that everything that is in either of the sets

A n B -> A intersect B which means that only the things that are in both of the sets

A c    -> A complement which means that everything in the object that is outside of A

A- B -> A minus B or A complement B which means everything in A except for anything in its overlap with B

- (AUB) -> not (A union B) which means everything outside of A and B

- (AnB) -> not (A intersect B) which means everything outer of the overlap of A and B



Set Theory Venn Diagrams Example Problem:

Mary asked 100 coffee drinkers whether they like cream or sugar in their coffee. 16 drinkers using cream only and 20 using both and 35 drinkers like sugar only.   How many like cream, sugar, cream but not sugar, sugar but not cream, cream and sugar, cream or sugar? Apply set theory Venn diagrams to solve.

Solution:We can find the solution for these questions using set theory Venn diagrams like

We should add 20 + 16 to find the no. of drinkers like cream

And 16+35 gives the number of sugar liking drinkers

Sugar but not cream is 35 and cream but not sugar is 16. Both gives 20 and we should add all 16+35+20 to find how many drinkers like cream or sugar which is 71.

Operations and Laws of Set Theory Venn Diagrams:

Operations on Sets:

Union of Sets
Intersection of sets
Disjoint of sets
Difference of two sets
Symmetrical difference of two sets
Complement of  a set
Algebraic laws:

Idempotent Laws
Identity laws
Associative laws
De-Morgan's Laws
Distributive laws