 # Operations on Sets – Union, Intersection, Difference, Cross Product | Set Operations Examples and Solutions

Sets are a collection of well-defined objects. If two are more sets are combined to form one set under the special conditions, then set operations are carried out. Students who want to know more about the Operations on Sets can check the below sections. On this page, we will provide detailed information like set operations definitions, examples, and Venn diagrams to understand them clearly.

## What are Set Operations?

Sets operations come into existence when two or more sets are joined to form one set under some conditions. Basically, we have 4 types of operations on sets. They are

• Union of Sets
• Intersection of Sets
• Complement of the Sets
• Cartesian Product of Sets

### Operations on Sets

Here we will discuss each of the sets operations in detail along with the examples.

1. Union of Sets

The union of two sets A and B is a set of elements that are in both A and. It is denoted by A U B.

So, A U B = { x | x ∈ A or x ∈ B }.

Examples:

If A = {1, 4, 8, 16, 32}, B = {3, 9, 27, 1, 6}

Now write every element of A and B in A U B without repetition.

A U B = {1, 3, 4, 6, 8, 9, 16, 27, 32} 2. Intersection of Sets

The intersection of two sets A and B means the set of elements that are common in both A and B. It is denoted by A ∩ B.

Therefore, A ∩ B = { x : x ∈ A and x ∈ B }.

Examples:

If A = {1, 4, 5, 10, 15, 8, 9}, B = {5, 10, 20, 25, 30}

Now we write the common elements from both sets A and B.

A ∩ B = {5, 10} 3. Difference or Complement of Set

when A and B are two sets, then their difference A – B means the elements of A but not the elements of B.

A minus B can be written as A – B.

A – B = {x : x ∈ A, and x ∉ B}

A – B never equal to B – A. i.e A – B ≠ B – A

If A and B are disjoint sets, then A – B = A and B – A = B.

Examples:

If A = {2, 4, 3, 6, 4, 8, 5, 10} and B = {1, 2, 3, 4}

Then, A – B includes elements of A but not elements of B.

A – B = {5, 6, 8, 10} Complement of a Set

The complement of a set is the set of elements that are not in that set. The complement of set a is denoted by A’.

Therefore, A’ = {x | x ∉ A }

A’ = (U – A) here U is the universal set that contains all elements.

Examples:

A = { x : x belongs to set of even integers }

U = {x : x belongs to set of integers}

A’ = U – A

So, A’ = {x : x belongs to set of odd integers} 4. Cartesian Product or Cross Product

The cartesian product of two non-empty sets A and B are denoted by A x B. The cross product is the set of all ordered pair of elements from A and B. The cartesian product is also known as the cross product.

A x B = { (a, b) | a ∈ A and b ∈ B }

The cross product of two sets A x B and B x A do not contain exactly the same ordered pairs.

So, A x B ≠ B x A.

Examples:

If A = {4, 5, 6} and B = (a, b}

The cross product of A and B have 6 ordered pairs.

A x B = {(4, a), (4, b), (5, a), (5, b), (6, a), (6, b)}

B x A = {(a, 4), (a, 5), (a, 6), (b, 4), (b, 5), (b, 6)}

Properties of Cartesian Product

• It is non-commutative. i.e A x B ≠ B x A
• A x B = B x A, when A = B
• A x B = { }, if either A = ∅ or B = ∅
• The cartesian product is non-associative. i.e (A x B) x C ≠ A x (B x C)
• Distributive property over intersection, union and set difference are
• A x (B U C) = (Ax B) U (A x C)
• A x (B ∩ C) = (A x B) ∩ (A x C)
• A x (B/C) = (A x B) / (Ax C)

### Set Operations and Examples

Question 1:

If A = {1, 3, 5, 7}, B = {2, 4, 6, 8}, C = {1, 2, 3, 4}, D = {5, 6, 7, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8} find

(i) A U B

(ii) A ∩ C

(iii) D’

Solution:

Given sets are

A = {1, 3, 5, 7}, B = {2, 4, 6, 8}, C = {1, 2, 3, 4}, D = {5, 6, 7, 8}, U = {1, 2, 3, 4, 5, 6, 7, 8}

(i) A U B = {1, 3, 5, 7} U {2, 4, 6, 8}

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

So, A U B = {1, 2, 3, 4, 5, 6, 7, 8}

(ii) A ∩ C = {1, 3, 5, 7} ∩ {1, 2, 3, 4}

= {1, 3}

So, A ∩ C = {1, 3}

(iii) D’ = U – D

= {1, 2, 3, 4, 5, 6, 7, 8} – {5, 6, 7, 8}

= {1, 2, 3, 4}

So, D’ = {1, 2, 3, 4}

Question 2:

If A = {10, 12, 15, 18}, B = {11, 15, 14, 16}, C = {15, 16, 18, 7} find

(i) A – B

(ii) B – A

(iii) A – C

Solution:

Given sets are

A = {10, 12, 15, 18}, B = {11, 15, 14, 16}, C = {15, 16, 18, 7}

(i) A – B = {10, 12, 15, 18} – {11, 15, 14, 16}

= {10, 12, 18}

So, A – B = {10, 12, 18}

(ii) B – A = {11, 15, 14, 16} – {10, 12, 15, 18}

= {11, 14, 16}

So, B – A = {11, 14, 16}

(iii) A – C = {10, 12, 15, 18} – {15, 16, 18, 7}

= {10, 12}

So, A – C = {10, 12}

Question 3:

If P = {a, b, d}, Q = {m, n, o}, R = {l, e, t, t, e, r} find

(i) P x Q

(ii) P x R

(iii) Q x R

Solution:

Given sets are

P = {a, b, d}, Q = {m, n, o}, R = {l, e, t, t, e, r}

(i) P x Q = {a, b, d} x {m, n, o}

= {(a, m), (a, n), (a, o), (b, m), (b, n), (b, o), (c, m), (c, n), (c, o)}

So, P x Q = {(a, m), (a, n), (a, o), (b, m), (b, n), (b, o), (c, m), (c, n), (c, o)}

(ii) P x R = {a, b, d} x {l, e, t, t, e, r}

= { (a, l), (a, e), (a, t), (a, r), (b, l), (b, e), (b, t), (b, r), (d, l), (d, e), (d, t), (d, r) }

So, P x R = { (a, l), (a, e), (a, t), (a, r), (b, l), (b, e), (b, t), (b, r), (d, l), (d, e), (d, t), (d, r) }

(iii) Q x R = {m, n, o} x {l, e, t, t, e, r}

= {(m l), (m, e), (m, t), (m, r), (n, l), (n, e), (n, t), (n, r), (o, l), (o, e), (o, t), (o, r)}

So, Q x R = {(m l), (m, e), (m, t), (m, r), (n, l), (n, e), (n, t), (n, r), (o, l), (o, e), (o, t), (o, r)}

### FAQs on Sets Operations

1. What are the 4 operations of sets?

The four basic operations on sets are the union of sets, the intersection of sets, set difference, and the cartesian product of sets. When two sets are combined under some constraints, then we use these set operations.

2. How can operations be performed on sets?

Based on the constraints when joining two sets, operations on sets are performed. Union means adding the elements of both sets, intersection means adding common elements from two sets, difference means adding the elements of the first set but not the second set. Cross product gives the ordered pairs, by taking the elements from both sets.

3. How to find A x B in sets?

A x B means the cross product of two sets A and B which means the set of ordered pairs (a, b) where a ∈ A, b ∈ B. The set builder form is A x B = { (a,b) |a ∈ A,b ∈ B }.

4. What are the properties of set operations?

The properties of set operations are commutative property, distributive property, associative property, identity property, idempotent, and complement.

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