OPERATIONS ON SETS - Unit 17 - Explanation Of Exercise 17.2 - Mathematics For Class X Science

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Unit 17: SETS AND FUNCTIONS
Explanation For Exercise 17.2

OPERATIONS ON SETS

• Union
• Intersection
• Difference
• Complement

Union Of Two Sets:
The union of two sets X and Y, denoted as XUY, is the set of all those elements which belong to X or to Y or to both X and Y.
i.e., XUY = {x | x ∈ X ∨ x ∈ Y}
For example:
If X = {1,3,5} and Y = {1,2,3,4} then XUY = {1,2,3,4,5}

Intersection Of Two Sets:
The intersection of two sets X and Y, denoted as X∩Y, is the set of all those elements which belong to both X and Y.
i.e., X∩Y = {x | x ∈ X ⋀ x ∈ Y}
For example:
If X = {2,4,6,8} and Y = {l,2,3,6} then X∩Y = {2,6}

Difference Of Two Sets:
For any two sets X and Y, the difference X — Y is the set of all the elements which belong to X but do not belong to Y. It is also denoted as X\Y.
i.e., X — Y = {x | x ∈ X ⋀ x ∉ Y}
Similarly, the difference Y — X is the set of all the elements which belong to Y but do not belong to X. It is also denoted as Y\X.
i.e., Y — X = {y | y ∈ Y ⋀ y ∉ X}
For example:
If X = {4,6,8,9,10} and Y = {2,4,6,8}
then X — Y ={ 9, 10}
and Y — X = {2}

Complement Of A Set:
If set A is a subset of universal set U then the complement of A is the set of all the elements of U which are not in A. It is denoted by A' or A^c.
Thus A' = U — A
i.e., A' = {x | x ∈ U ⋀ x ∉ A}
For example:
If U ={1,2,3,...,20} and A = {1,3,5,...,19}
then A' ={2,4,6,...,20}

SYMMETRIC DIFFERENCE OF TWO SETS

The symmetric difference of two sets A and B, denoted AΔB, is the set of all the elements of A or B which are not common in both the sets.
i.e., AΔB = {x | x ∈ AUB ⋀ x ∉ A∩B}
or AΔB = (AUB) — (A∩B)
For example:
If A ={1,2,3,4,5} and B ={1,3,5,7}
then AΔB = {2,4,7}

FURTHER TYPES OF SETS:

are given below:
Disjoint Sets:
Two sets A and B are called disjoint sets if they have no element common.
i.e., Two sets A and B are disjoint sets if A∩B = ∅
For example:
The sets A ={1,3,5} and B = {2,4,6} are disjoint sets.

Overlapping Sets:
Two sets A and B are called overlapping sets if there is at least one element common in both. Moreover neither of them is subset of other.
i.e., Two sets A and B are overlapping sets if A∩B ≠ ∅ and A⊈B or B⊈A
For example:
The sets A = {1,2,3,4} and B ={2,4,6,8} are overlapping sets.

Exhaustive Sets:
If two sets A and B are subsets of universal set U then A and B are called exhaustive sets if AUB= U.
For example:
If A ={1,2,3,4}, B = {1,3,5} and U = {1,2,3,4,5}
then A and B are exhaustive sets because AUB = U

Cells:
If A and B are two non-empty subsets of universal set U then A and B are called cells if they are disjoint as well as exhaustive sets.
i.e., A and B are called cells if A and B are non-empty subsets of U and
A∩B = ∅
Also AUB = U
For example:
If A = {1,3,5,7,9}, B = {2,4,6,8,10} and U = { 1,2,3,...,10) then A and B are cells.

LAWS RELATED TO OPERATIONS ON SETS

Some important laws related to the operations on sets are as under:
Identity Laws:
For any set A

• AU∅ = A
• AUU = U
• A∩U = A
• A∩∅ = ∅

Idempotent Laws:
For any set A

• AUA = A
• A∩A =A

Laws of the Complement:
For any set A

• AUA' = U
• A∩A' = ∅
• (A') = A
• U' = ∅ and ∅' = U

De Morgan's Laws:
For any two sets A and B:

• (AUB)' =A'∩B'
• (A∩B)' = A'UB'