Maths Set Theory

Operations On Sets

Set Theory - Operations On Sets
Written by Mary Christina

OPERATIONS ON SETS:

Union:

Let X and Y be two sets. We define the union of these sets as follows
X∪Y= { z/z∈X or z∈Y}
Here we see that X∪Y contains all the elements of X and all elements of Y.

Venn Diagram for Union of sets:

Set Theory - Operations on Sets, Venn Diagram for Union of Sets

Example:
Let A={ a,b,c,d} and B={ a,c,e}. Then A∪B is given by
A∪B={a,b,c,d,e}

Intersection:

Let X and Y be two sets. We define the intersection of these sets as follows
X∩Y={z/z∈X and z∈Y}
Here we see that X∩Y contains only the elements that are contained in both X and Y.
Note: X∩Y ⊆X and also X∩Y ⊆Y

Venn diagram for Intersection of sets:

Set Theory - Operations on Sets, Venn Diagram for Intersection of Sets

Example:
Let A={1,2,3,4,5} and B={3,4,5,6}.Then A∩B is given by
A∩B={3,4,5}

Set Difference:

Let X and Y be two sets. We define the set difference as follows
X\Y={z/z∈X but z∉Y}
Here we see that X\Y contains only elements of X but does not contain the elements of Y.
Some authors use X-Y to denote set difference but most commonly used notation for set difference is X\Y.

Venn diagram for set difference:

Set Theory - Operations on Sets, Venn Diagram for Set Difference

Example:
Let A={5,10,15,20} and B={6,10,12,18,24}.Then A\B is given by
A\B={5,15,20}

Symmetric difference:

Let X and Y be two sets. we define the symmetric difference of these sets
x ΔY=(X\Y) ∪ (Y\X).
XΔY contains all elements in X∪Y but does not contain the elements in X∩Y.

Venn diagram for the symmetric difference:

Set Theory - Operations on Sets, Venn Diagram for Symmetric Difference

Example:
Let A={ x\x is a positive integer less than 12} i.e. A={1,2,3,4,5,6,7,8,9,10,11} and B={ 1,2,4,6,7,8,12,15}
Then AΔB is given by
AΔB={3,5,9,10,11,12,15}

Complement:

Let U be the universal set. and let X be a subset of U i.e. X⊆U then U\X is called the complement of the set X with respect to U. We denote U\X by X’ and is called the complement of X.
The set difference X\Y can be viewed as the complement of Y with respect to X.

Venn Diagram for complement of set:

Set Theory - Operations on Sets, Venn Diagram for Complement of Set

Example:
Let U={ 1,2,3,4,5,6,7,8,9,10} and A={2,4,6,8,10} Then A’ is given by
A’={ 1,3,5,7,9}.

Disjoint Sets:

Let X and Y be two sets. These two sets X and Y are said to be disjoint if they does not have elements in common. i.e. X∩Y={}

Venn Diagram for Disjoint sets:

Set Theory - Operations on Sets, Venn Diagram for Disjoint Sets

Example:
Let A={a,b,c,d} and B={p,q,r}
Here we see that A and B don’t have elements in common. thus A and B are disjoint sets.

About the author

Mary Christina

Mary Christina from Chennai, India. She has completed Master Degree in Mathematics and she loves to teach and share about her knowledge in maths for High School and College Students...