ICSE Class 8 Maths Sets Chapter 02 Operations on Sets

Operations On Sets

Union of Sets

Intersection of Sets

Difference of Sets

Distributive Laws

De Morgan's Laws

Cardinal Number of Sets

Introduction

This chapter deals with various operations on sets and the results of those operations. Later, some important laws, namely, distributive law and De Morgan's law, have been stated.

Union of Sets

The union of sets A and B is a set of all the elements that are either in set A or in set B or in both sets A and B. It is represented by the symbol ∪.

If x is an element in A ∪ B, then x ∈ A or x ∈ B or x belongs to A as well as B.

Example 1: If A = {x | x = 2a, a ≤ 5, a ∈ N}, B = {x | x = 3a, a ≤ 5, a ∈ N}, and C = {x | x = 6a, a ≤ 5, a ∈ N}, find A ∪ B ∪ C.

In Roster form,

A = {2, 4, 6, 8, 10}, B = {3, 6, 9, 12, 15}, and

C = {6, 12, 18, 24, 30}

The union of the sets consists all the elements in the sets without repetition of any.

A ∪ B = {2, 3, 4, 6, 8, 9, 10, 12, 15}

B ∪ C = {3, 6, 9, 12, 15, 18, 24, 30}

A ∪ C = {2, 4, 6, 8, 10, 12, 18, 24, 30}

A ∪ B ∪ C = {2, 3, 4, 6, 8, 9, 10, 12, 15, 18, 24, 30}

Union Facts

A ∪ B = B ∪ A

(A ∪ B) ∪ C = A ∪ (B ∪ C)

A ∪ A = A

A ∪ ∅ = A

A ∪ ξ = ξ

A ∪ A' = ξ

If A ⊆ B, then A ∪ B = B.

If A ∪ B = ∅, then A = ∅ and B = ∅.

Try this!

If A = {1, 2, 3} and B = {3, 4, 5} find A ∪ B.

Intersection of Sets

The intersection of sets A and B is a set containing all the elements that are common to set A and set B. It is represented by the symbol ∩.

If x is an element in A ∩ B, then x belongs to A as well as B.

Example 2: If A = {x | x = \(\frac{30}{a}\), a ∈ N and x ∈ N}, B = {x | x = \(\frac{45}{a}\), a ∈ N, x ∈ N}, and C = {x | x = \(\frac{90}{a}\), a ∈ N, x ∈ N}, find A ∩ B ∩ C.

Teacher's Note

Understanding union and intersection helps us organize information in real life, such as finding common interests between friend groups or combining different categories of data.

Intersection Facts

A ∩ B = B ∩ A

(A ∩ B) ∩ C = A ∩ (B ∩ C)

A ∩ A = A

A ∩ ∅ = ∅

A ∩ ξ = A

A ∩ A' = ∅

If A ⊆ B, then A ∩ B = A.

If A ∩ B = ∅, then A and B are disjoint sets.

Try this!

If A = {1, 2, 5, 8} and B = {5, 6, 7, 8} find A ∩ B.

In Roster form,

A = {1, 2, 3, 5, 6, 10, 15, 30}

B = {1, 3, 5, 9, 15, 45}

C = {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}

The intersection of the given sets will contain the underlined common factors.

A ∩ B ∩ C = {1, 3, 5, 15}

Teacher's Note

Intersection represents finding common ground, like when different groups need to identify shared members or when filtering data for multiple criteria.

Difference of Sets

The difference of set A and set B is a set containing all those elements in set A that do not belong to set B.

If x is an element in A - B, then x ∈ A, but x ∉ B.

Example 3: If A = {x | x = \(\frac{18}{a}\), a ∈ N and x ∈ N} and B = {x | x = \(\frac{24}{a}\), a ∈ N and x ∈ N}, find A - B and B - A.

In Roster form, A = {1, 2, 3, 6, 9, 18} and B = {1, 2, 3, 4, 6, 8, 12, 24}

The difference of A and B will have all the elements in A that are not in B.

As A ∩ B = {1, 2, 3, 6}, A - B = {9, 18}.

The difference of B and A will have all the elements in B that are not in A.

As B ∩ A = {1, 2, 3, 6}, B - A = {4, 8, 12, 24}.

Difference Fact

A - B ≠ B - A

Try this!

If A = {1, 3, 10, 15} and B = {2, 4, 2, 8, 10}, find A - B and B - A.

Teacher's Note

Difference of sets helps identify unique elements, like finding items in one person's collection that another person doesn't have.

Laws of Union and Intersection of Sets

Distributive Laws

Consider A = {a, b, c, d, e}, B = {c, d, e, f, g}, C = {a, c, e, g, h}

(i) B ∩ C = {c, e, g},

A ∪ (B ∩ C) = {a, b, c, d, e, g},

A ∪ B = {a, b, c, d, e, f, g} and

A ∪ C = {a, b, c, d, e, g, h}

Thus (A ∪ B) ∩ (A ∪ C) = {a, b, c, d, e, g}(2)

Combining (1) and (2), we obtain

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

(ii) B ∪ C = {a, c, d, e, f, g, h},

A ∩ (B ∪ C) = {a, c, d, e}

A ∩ B = {c, d, e} and A ∩ C = {a, c, e}

Thus (A ∩ B) ∪ (A ∩ C) = {a, c, d, e}(2)

Combining (1) and (2), we obtain

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

De Morgan's Laws

Consider A = {a, b, c, d, e}, B = {c, d, e, f, g}, C = {a, c, e, g, h}, ξ = {a, b, c, d, e, f, g, h}

Then A' = {f, g, h}, B' = {a, b, h}, and C' = {b, d, f}

The complement of unions is the intersection of complements.

(i) A ∪ B = {a, b, c, d, e, f, g} and (A ∪ B)' = {h}

A' ∩ B' = {h}

Thus (A ∪ B)' = A' ∩ B'

A ∪ B ∪ C = {a, b, c, d, e, f, g, h} and (A ∪ B ∪ C)' = ∅

A' ∩ B' ∩ C' = ∅

Thus (A ∪ B ∪ C)' = A' ∩ B' ∩ C'

The complement of intersections is the union of complements.

(ii) A ∩ B = {c, d, e} and (A ∩ B)' = {a, b, f, g, h}

A' ∪ B' = {a, b, f, g, h}

Thus (A ∩ B)' = A' ∪ B'

A ∩ B ∩ C = {c, e} and (A ∩ B ∩ C)' = {a, b, d, f, g, h}

A' ∪ B' ∪ C' = {a, b, d, f, g, h}

Thus (A ∩ B ∩ C)' = A' ∪ B' ∪ C'

Cardinal Number of Union and Intersection of Sets

1. n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Example 4: Let set A = {3, 4, 5, 6, 7}, set B = {1, 2, 3, 4}, n(A) = 5, n(B) = 4

Thus, we observe that the sum of cardinal numbers of sets A and B = n(A) + n(B) = 5 + 4 = 9

The number of common elements or n(A ∩ B) = 2

Thus, n(A) + n(B) - n(A ∩ B) = 5 + 4 - 2 = 7 = n(A ∪ B)

CHECK: A ∪ B = {1, 2, 3, 4, 5, 6, 7}, n(A ∪ B) = 7

2. n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

Example 5: Let set A = {5, 6, 7, 8, 9, 10}, set B = {7, 8, 9, 10, 11}, n(A) = 6, n(B) = 5.

Thus, we observe that the sum of cardinal numbers of sets A and B = n(A) + n(B) = 6 + 5 = 11

The numbers of elements in the union of set A and set B or n(A ∪ B) = 7 [∴ A ∪ B = {5, 6, 7, 8, 9, 10, 11}]

Thus, n(A) + n(B) - n(A ∪ B) = 6 + 5 - 7 = 4 = n(A ∩ B)

CHECK: A ∩ B = {7, 8, 9, 10}, n(A ∩ B) = 4

Remember

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(A ∪ B)' = A' ∩ B'

(A ∪ B ∪ C)' = A' ∩ B' ∩ C'

(A ∩ B)' = A' ∪ B'

(A ∩ B ∩ C)' = A' ∪ B' ∪ C'

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

Teacher's Note

Cardinal numbers of sets help us count and analyze data, similar to how retailers track inventory or how social scientists analyze survey responses.

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