Set Notation and Venn Diagrams

​Venn Diagrams and Sample Space

​Learning objective

• Correctly draw and label a Venn Diagram 

• Understand set notation (union, intersection, A, A’, etc)

Set

A set is a list of objects in no particular order; they could be numbers, letters or even words.

Sets are named and represented using capital letters.

eg A = {1, 2, 3, 4, 5}

Elements of a Set

The items present in a set are called either elements or members of a set. The elements of a set are enclosed in curly brackets separated by commas.

To denote that an element is contained in a set, the symbol '∈' is used. eg 2 ∈ A.

If an element is not a member of a set, then it is denoted using the symbol '∉'. eg 9 ∉ A.

• ​Cardinal Number of a Set

The cardinal number(also known as order of a set) denotes the total number of elements in the set.

For example , For natural even numbers less than 10, n(A) = 4

Sets Notations

T​here are different set notations used for the representation of sets in set theory. They differ in the way in which the elements are listed. The three set notations used for representing sets are:

• Semantic form

• Roster form

• Set builder form

​Sets Symbols

Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.

​Universal Set

A universal set is the collection of all the elements regarding a particular subject. The universal set is denoted by the letter 'U'. Example: Let U = {The list of all road transport vehicles}.

Empty or Null Sets

A set that does not contain any element is called an empty set or a null set. An empty set is denoted using the symbol '∅' or '{ }'.

Union of Sets

Union of sets, which is denoted as A U B, lists the elements in set A and set B or the combination all elements in set A and set B. For example, {1, 3} ∪ {1, 4} = {1, 3, 4}

Intersection of Sets

The intersection of sets which is denoted by A ∩ B lists the elements that are common to both set A and set B. For example, {1, 2} ∩ {2, 4} = {2}

Venn Diagram

A Venn diagram is a way of representing sets visually.

Lets Consider the whole numbers from 1 to 10.
We will define two sets taken from this group of numbers:

Set A = the odd numbers in the group = { 1 , 3 , 5 , 7 , 9 }
Set B = the numbers which are 6 or more in the group = { 6 , 7 , 8 , 9 , 10 }

The Following Diagram is a venn Diagram showing the information stated above

The two large circles represent the two sets(A and B).

The numbers which appear in both sets are 7 and 9. These will go in the central section, because this is part of both circles(Union).

The numbers 1, 3 and 5 still need to be put in Set A, but not in Set B, so these go in the left section of the diagram(set A only).

Similarly, the numbers 6, 8 and 10 are in Set B, but not in Set A, so will go in the right section of the diagram(set B only).

The numbers 2 and 4 are not in either set, so will go outside the two circles .

​Please watching the following video

https://www.youtube.com/watch?v=xZELQc11ACY

​Venn Diagram

A venn diagram is usually represented by a rectangle that contains all the elements that are being considered in the problem.

The universal set is usually labelled with the symbol ξ or μ

Complement of Set

The region outside the circle represents the complement

A^c = {0, 2, 4, 6, 8, 10}.

Commonly sets interact. For example, you and a friend decide to have a party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.​

​

1. to show these comparisons

​The intersection of two sets contains only the elements that are in both sets. The intersection is notated A ⋂ B.

Multiple Choice

What is the intersection of sets A and B?

A ⋂ B = ______

• 1, 2, 3, 4. 5

• 1, 2

• 3, 4, 6, 8

• 3, 4

Multiple Choice

Which colour is the intersection?

• blue

• green

• yellow

• red

​The union of two sets contains all the elements contained in either set (or both sets). The union is notated A ⋃ B.

Multiple Choice

What numbers or elements is complement of set A (A')

• 1, 3, 5

• 2, 4, 6

• 1, 2, 3, 4, 5, 6,

• 2, 4

Multiple Choice

A ' is ___________________

• Pentagon

  Hexagon

  octagon

• Nanogon 

  Heptagon

• Heptagon

• Pentagon

  Nanogon

  Hexagon

  Octagon

  Heptogon

Multiple Choice

Let the universal set, U = {a, e, i, o, u} and set A = {a, e}, then complement of set A is:

• {i, o, u}

• {a. i}

• {a. o}

• {a, u}

Multiple Choice

The complement of set A is written as?

• A'

• A

• A''

• A!

Multiple Choice

The Universal Set = { -4, 3, -2, -1, 0, 1, 2, 3 ,4} and A = {0}. What is the complement of A?

• {-4, -3, -2, -1, 0, 1, 2, 3}

• {-3, -2, -1, 1, 2, 3}

• {-4, -3, -2, -1, 1, 2, 3, 4}

• {-4, -3, -2, -1, 1, 2, 3}

Multiple Choice

If U = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and A = {12, 13, 14, 20}, what is the complement of A?

• {11, 15, 16, 17, 18, 19}

• {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

• {12, 13, 14, 20}

• {15, 16, 17, 18, 19}