Operation on Sets

Sets in maths are collections of distinct objects, and through these sets, we can carry out various operations to join, compare, or manipulate the sets. Knowing these types of operations is crucial in mathematics since it represents the base from which most complex subjects, including probability, algebra, and logic, spring forth.  

1.0What are Sets?

A set is defined as a collection of unique objects called elements. The elements can be numbers, letters, or even things, and sets are typically denoted using curly braces {}. For Example, {Red, Blue, Pink, Purple} or {2,4,6,8,10}.

2.0What are Set operations?

The set operations are those that combine or relate sets, such as union, intersection, difference, and complement. Such operations result in new sets based on the relations among the originals. Operation on sets examples are the Union of sets, the intersection of sets, the difference of sets, etc. 

3.0Types of Set Operations

1. Union of Sets (∪)

The union of two sets combines all elements from both sets, excluding duplicates.

Example: Let’s consider two sets:

• A = {1, 2, 3}
• B = {3, 4, 5}

The union of A and B will be:

• A ∪ B = {1, 2, 3, 4, 5}

2. Intersection of Sets (∩)

The intersection of two sets consists only of the elements common to both sets.

Example: With the sets:

• A = {1, 2, 3}
• B = {3, 4, 5}

The intersection of A and B will be:

• A ∩ B = {3}

If there are no common elements, the intersection is the empty set (denoted by ∅).

3. Difference of Sets (−)

The difference between two sets refers to elements in one set, not the other.

Example: Using sets:

• A = {1, 2, 3}
• B = {3, 4, 5}

The difference A − B will be:

• A − B = {1, 2}

Similarly, B − A will be:

• B − A = {4, 5}

4. Complement of a Set (′)

The complement of a set contains all elements from the universal set that do not belong to the given set.

Example: Let U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6}. The complement of A is:

• A′ = {1, 3, 5}

5. Symmetric Difference (△)

The symmetric difference between two sets contains elements in either set but not both.

Example: With sets:

• A = {1, 2, 3}
• B = {3, 4, 5}

The symmetric difference will be:

• A △ B = {1, 2, 4, 5}

6. Subset ($\subseteq$)

If two Sets A and B are given such that all the elements of A are in set B, Set A will be known as the Subset of B. 

Example: 

• Let A = {1, 2} and B = {1, 2, 3, 4}
• Here, every element of A (1 and 2) is also in set B; it can be mathematically represented as A $\subseteq$B

7. Power Set [P(A)]

A power set is the set of all subsets of that set, including the empty set and the set itself.

Example: 

• Let Set A = {1, 2} 
• The power set of A is 

P(A) = {$\varphi$, {1}, {2}, {1,2}}

8. Venn diagrams are used for the visual representation of sets. 

The Union of sets(AUB) is represented by the entire area covered by both circles, including the common area.

• The overlapping area indicates the intersection of sets (A∩B). 
• The difference is the non-overlapping part of one circle.
• Complement is anything outside the circle of a set but within the universal set.

(refer to the below diagrams for more clarity)

4.0Properties of Set Operations

1. Commutative Property:

The order in which we combine sets for union and intersection does not matter: 

AUB=BUA and A∩B=B∩A

2. Associative Property:

The grouping of sets for union and intersection does not change the result: 

(AUB)UC=AU(BUC) and (A∩B)∩C=A∩(B∩C)

3. Distributive Property:

Union distributes over the intersection and vice versa: 

AU(B∩C)=(AUB)∩(AUC) and A∩(BUC)=(A∩B)U(A∩C)

4. Identity Property:

The union of any set with the empty set is the set itself: 

A$U\varphi$=A

The intersection of any set with the universal set is the set itself: 

A∩U=A

5. Complement Laws:

A set A and its complement A' are described by the following relations:

AUA'=U and A∩A'= $\varphi$

5.0Solved Examples 

Problem 1: Given the following sets:

• A = {1, 2, 3, 4, 5}
• B = {4, 5, 6, 7, 8}
• C = {5, 6, 7, 8, 9}

Find the following:

a) A ∪ B
b) A ∩ B
c) (A ∪ B) ∩ C
d) (A ∩ B) ∪ C
e) A − (B ∪ C)

Solution: 

a) AUB={1,2,3,4,5}{4,5,6,7,8}={1,2,3,4,5,6,7,8}

b)A∩B={1,2,3,4,5}{4,5,6,7,8}={4,5}

c) (AUB)∩C={1,2,3,4,5,6,7,8}{5,6,7,8,9}={5,6,7,8}

d)(A∩B)UC={4,5}{5,6,7,8,9}={4,5,6,7,8,9}

e)A-(BUC)= {1,2,3,4,5}-{4,5,6,7,8}{5,6,7,8,9}={1,2,3,4,5}-{4,5,6,7,8,9}={1,2,3}

Problem 2: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set, and let A = {2, 4, 6, 8} and B = {3, 6, 9}.

Find the following:

a) A′ (complement of A)
b) (A ∪ B)′ (complement of the union of A and B)
c) (A ∩ B)′ (complement of the intersection of A and B)

Solution: 

a)A'=U-A={1,2,3,4,5,6,7,8,9,10}-{2,4,6,8}={1,3,5,7,9,10}

b)(A ∪ B)′=U-(A ∪ B)={1,2,3,4,5,6,7,8,9,10}-{2,4,6,8}{3,6,9}

={1,2,3,4,5,6,7,8,9,10}-{2,3,4,6,9}={1,5,7,8,10}

c)(A∩B)'=U-(A∩B)={1,2,3,4,5,6,7,8,9,10}-{2,4,6,8}{3,6,9}

={1,2,3,4,5,6,7,8,9,10}-{6}={1,2,3,4,5,7,8,9,10}

Problem 3: In a survey of 100 students, 70 students like English (E), 60 students like History (H), and 40 students like both English and History. How many students like either English or History but not both?

Solution: No. of students who like English P(E) = 70, 

No. of students who like History P(H) = 60, 

No. of students who like both English and history P(EH)=40, 

No. of students who like both the subject 

P(EUH)=P(E)+P(H)-P(E∩H)

P(EUH)=70+60-40=90

No. of students who like either English or history but not both

 P(EUH)-P(E∩H)=90-40=50

A Venn diagram like this can also solve this 

No. of students who only like English = 30

No. of students who only like History = 20

No. of students who like either English or history but not both = 50

No. of students who neither like English nor history = 10