# Operations on Sets

Now that we have an idea about what a set is, we can move on to understand how to work with them. Is there a way to merge two sets together? What is the intersection of sets and how do we find the intersection of two sets? Should we even be allowed to add two sets together? Here we will find out all the answers!

## Operations on Sets

Unlike the real world operations, mathematical operations do not require a separate no-contamination room, surgical gloves, and masks. But certainly, expertise to solve the problem, special tools, techniques, and tricks as well as knowledge of all the basic concepts are required to obtain a solution. Following are some of the operations that are performed on the sets: –

• Union
• Intersection
• Difference
• Complement

Let’s deal with them one by one.

## Union of Sets

Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Then, A U B is represented as the set containing all the elements that belong to both the sets individually. Mathematically,

A U B = {x : x ϵ A or x ϵ B}

So, A U B = {2, 4, 6, 8, 10, 12},

here the common elements are not repeated.

### Properties of (A U B)

• Commutative law holds true as (A U B) = (B U A)
• Associative law also holds true as (A U B) U {C} = {A} U (B U C)

Let A = {1, 2} B = {3, 4} and C = {5, 6}
A U B = {1, 2, 3, 4} and (A U B) U C = {1, 2, 3, 4, 5, 6}
B U C = {3, 4, 5, 6} and A U (B U C) = {1, 2, 3, 4, 5, 6}
Thus, the law holds true and is verified.

• A U φ = A (Law of identity element)
• Idempotent Law – A U A = A
• Law of the Universal set (U): (A U U) = U

## Intersection of Sets

An intersection is the collection of all the elements that are common to all the sets under consideration. Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then A ∩ B or “A intersection B” is given by:

“A intersection B” or A ∩ B = {6, 8}

Mathematically, A ∩ B = {x : x ϵ A and x ϵ B}

### Properties of the Intersection – A ∩ B

The intersection of the sets has the following properties:

• Commutative law – A ∩ B = B∩ A
• Associative law – (A ∩ B)∩ C = A ∩ (B∩ C)
• φ ∩ A = φ
• U ∩ A = A
• A∩ A = A; Idempotent law.
• Distributive law – A ∩ (BU C) = (A ∩ B) U (A ∩ C)

## Difference of Sets

The difference of set A and B is represented as:

A – B = {x : x ϵ A and x ϵ B}

Conversely, B – A = {x : x ϵ A and x ϵ B}

Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5} and B – A = {8}. The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets; it means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set.

## Complement of Sets

If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as A’) will contain ALL the elements which belong to the Universal set U but NOT to set A.

Mathematically, A’ = U – A

Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A.

### Properties of Complement Sets

• A U A’ = U
• A ∩ A’ = φ
• De Morgan’s Law – (A U B)’ = A’ ∩ B’ OR (A ∩ B)’ = A’ U B’
• Law of double complementation : (A’)’ = A
• φ’ = U
• U’ = φ

Hence, these are the basic concepts and operations on Sets.

## Solved Examples For You

Question 1: Let A = {1, 3, 5, 7} B = {5, 7, 9, 11} and C = {1, 3, 5, 7, 9, 11, 13} prove that:

(A∩ B) U (A ∩ C) = A ∩ (BU C)

Answer : B U C = {1, 3, 5, 7, 9, 11, 13}
A ∩ (B U C) = {1, 3, 5, 7}
Hence, A ∩ B = {5, 7}
A ∩ C = {1, 3, 5, 7}
(A∩ B) U (A∩ C) = {1, 3, 5, 7}                    … {Hence proved}

Question 2: Prove De Morgan’s Law.

Answer : De Morgan’s law is a very important and crucial concept in Set Theory. It serves numerous applications in the real world regarding Boolean algebra. The statement of the law reads as –

(A U B)’ = A’ ∩ B’       OR      (A ∩ B)’ = A’ U B’

We will prove this law by separately dealing with both the statements.

### Case I: (A U B)’ = A’ ∩ B’

Let A = {Set of natural numbers ≤ 10} and B = {Even numbers ≤ 10}
So, A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 4, 6, 8, 10}
A U B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(A U B)’ = φ               … {Equation 1}
A’ = φ and B’ = {1, 3, 5, 7, 9}
A’ ∩ B’ = φ                … {Equation 2}
By Equation 1 and 2 – L.H.S. = R.H.S.

### Case II: (A ∩ B)’ = A’ U B’

Taking the same example, i.e. A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 4, 6, 8, 10}
A ∩ B = {2, 4, 6, 8, 10}
(A∩ B)’ = {1, 3, 5, 7, 9}        … {Equation 3}
A’ = φ
B’ = {1, 3, 5, 7, 9}
A’ U B’ = {1, 3, 5, 7, 9}         … {Equation 4}
By Equation 3 and 4 – L.H.S. = R.H.S.

Question 3: What does ∩ mean?

Answer: In mathematics, the intersection of two given sets is the largest set that contains all the elements that are common to both the sets. In addition, the symbol for denoting intersection of sets is ∩, which is a common representation of sets.

Question 4: State the symbols of sets?

Answer: Basically there are four types of sets namely:

• Subset- A ⊆ B, where A is a subset of B and set A is part of set B.
• Strict Subset/ Proper subset- A⊂ B, where A is a subset of B, but A is not equal to B.
• Not Subset- A ⊄ B, here set A is not a subset of B.
• Superset- A ⊇ B, where A is a superset of B and set A includes set B.

Question 5: What is the intersection of two sets?

Answer: When two sets intersect they form a new set that contains all the elements of both the sets. Furthermore, we can write this intersection as $$A \cap B$$.

Question 6: What is the intersection of an empty set?

Answer: It is a set with no elements. Also, if there are no elements in at least one of the sets that we are trying to find then the two sets have no elements in common.

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