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Operations on Sets

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Operations on Sets - Lesson Summary

Union of sets
 
Let P and Q be two sets.

P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9}

The union of sets P and Q is a set that consists of the elements from both the sets, P and Q.

P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9}
 
Set P union Q consists of all the elements of sets P and Q.
 
Union: The union of two sets A and B is set C, which consists of all elements that are either in A or in B.
 
Symbolically, the union of sets A and B is represented as A ∪ B = {x: x ∈ A or x ∈ B}.
 
The Venn diagram representation of two sets A and B is as shown.
 

 
The region shaded in green represents the union of the sets, A and B. Both the circles are shaded since set A union B consists of all the elements of sets A and B.

Properties exhibited by the union of two sets:
(i) A ∪ B = B ∪ A (Commutative law)
(ii) ( A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law)
(iii) A  ∪ ∅ = A (Law of identity element, ϕ is the identity of ∪)
(iv) A ∪ A = A (Idempotent law)
(v) U ∪ B = U(Law of U)
 
Intersection of sets
 
Consider the sets P and Q,

P = {1, 2, 3, 4, 5}, Q = {5, 6, 7, 8, 9}

In sets P and Q, the common element is five.
Therefore, the intersection of sets P and Q is 5.
 
P ∩ Q = {5}
 
Intersection: The intersection of two sets A and B is the set of all those elements that belong to both A and B.
 
Symbolically, the intersection of sets A and B is represented as A ∩ B = {x: x ∈ A and x ∈ B}.
 
The Venn diagram representation of intersection of two sets A and B is as shown.
 

 
The region shaded in green represents the common elements belonging to the two sets.
 
Property of the intersection of sets
 
Consider two sets C and D
 
C = {2, 4, 6, 8}, D = {3, 5, 7, 9}
 
There is no common element among the sets, C and D.
Therefore, C intersection D is a null set.
 
C ∩ D = Ø
 
The Venn diagram representation of sets C and D is as shown.
 

 
Properties of the intersection of two sets
 
1. A ∩ B = B ∩ A (Commutative law)
2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law)
3. ∅ ∩ A = ∅ , U ∩ A = A (Law of φ and U)
4. A ∩ A = A (Idempotent law)
5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law)
 
Distributive law can be verified with the help of Venn diagrams:
 
LHS of the equation A ∩ (B ∪ C) can be represented as shown below:
 
 
 
 
RHS of the equation:
 
(A ∩ B) can be represented as:
 

 
(A ∩ C) can be represented as:
 

 
(A ∩ B) ∪ (A ∩ C) can be represented as
 

 
Difference of two sets

Consider two sets X and Y.

X = {2, 3, 4, 5, 6} Y = {5, 6, 7, 8}

Difference of sets X and Y taken in the same order ( X - Y), contains the elements that are present in X but not in Y.
 
Difference of sets Y and X taken in the same order ( Y - X), contains the elements that are present in Y but not in X.
 
It can be observed that, X - Y ≠ Y - X.
 
Representation of set X - Y through a Venn diagram:
 

 
Only the region of set X, which is not shared by set Y, is shaded.
 
Representation of set Y - X through a Venn diagram:
 

 
Only the region of set Y that is not shared by set X is shaded.
 
The difference of any two sets A and B taken in order is represented symbolically as:
A – B = {x: x ∈ A and x ∉ B}

Similarly, B minus A is written symbolically as:
B – A = {x: x ∈ B and x ∉ A}
 
Complement of a set
 
Complement of a set: Let U be the universal set and A be a subset of U. Then the complement of A is the set of all elements of U that are not the elements of A.
 
The complement of a set A is symbolically represented as: A' = {x: x ∈ U and x ∉ A}.
 
The Venn diagram representation of the complement of a set is as shown.
 

 
The region shaded in green represents the complement of set A.
 
The complement of a set depends on the universal set.
 
Ex: The complement of the set of natural numbers with the universal set being the set of integers.
 
N = {1,2,3,....}

Z = {.., -2, -1, 0 , 1, 2,....}
 
N' = {.., -2, -1, 0} 

Observations on complements of sets:
 
If A is a subset of the universal set U, then its complement A′ is also a subset of U.
The complement of the union of two sets is the intersection of their complements, and the complement of the intersection of two sets is the union of their complements.
 
DeMorgan’s law:
(A ∪ B) = A' ∪ B'

and

(A ∩ B) = A'  B'

Complement laws:
(i) A ∪ A' = U (ii) A ∩ A' = ∅

Law of double complementation: (A' )' = A

Laws of empty set and universal set:   ∅' = U and U' = ∅.

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