Intersection of Two Sets

Set is a fundamental part of the mathematics. This set concept is applied in every branch of mathematics. Sets are used in relations and functions. The application of sets are geometry,  sequences, Probability etc. Sets are used in everyday life such as a volleyball team, vowels in alphabets, various kinds of geometry shapes etc.

There are two main important operations in sets. They are  union and intersection of two sets. Let us learn the concepts and properties of intersection of two sets. We will learn some example problems and give practical problems about intersection of two sets.

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Intersection of Sets:

Let A and B be any two sets.  A intersection B is the set of all elements which are similar to both A and B. The symbol `nn` is used to denote the intersection. A intersection B is the set of all those elements which belong to both A and B. Symbolically, we write A `nn` B = { x : x `in` A and x `in` B }.

Ex:

Consider the two sets X = { 1, 5, 8, 9 } and Y = { 5, 6, 9, 15 } . Find X intersection Y.

Sol:

We see that 5, 9 are the only elements which are similar to both X and Y. Hence  X ∩ Y = { 5, 9 }

Some Properties of Operation of Intersection:

(i) Commutative law:  A ∩ B = B ∩ A

(ii) Associative law: ( A ∩ B ) ∩ C = A ∩ ( B ∩ C )

(iii) Law of identity: φ ∩ A = φ, U ∩ A = A (Law of φ and U).

(iv) Idempotent law: A ∩ A = A

(v) Distributive law: A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) i. e., ∩ distributes over ∪

Between, if you have problem on these topics union of sets, please browse expert math related websites for more help on Union Set.

Practice Problems on Intersection of Two Sets:

Problem 1:

Find the intersection of each of the following two sets:

1. X = { 1, 3, 5 }   Y = { 1, 2, 3 }

2. A = [ a, e, i, o, u ]   B = { a, b, c }

3. A = { 1 , 2 , 3 }    B = `Phi`

Sol:

1. X `nn` Y = { 1, 3 }

2. A `nn` B = [ a ]

3. A `nn` B = `Phi`

Problem 2:

If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find

(i) A ∩ B  (ii) B ∩ C (iii) A ∩ C ∩ D

(iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)

(vii) A ∩ D (viii) A ∩ (B ∪ D)

Sol:

i) A ∩ B = { 7, 9, 11 }

ii) B ∩ C = { 11, 13 }

iii) A ∩ C ∩ D = Nill

iv) A ∩ C = { 11 }

v) B ∩ D = nill

vi) A ∩ (B ∪ C) = { 7, 9, 11, 13 }

vii) A ∩ D = nill

(viii) A ∩ (B ∪ D) = { 7, 9, 11}