Set Relation and Function

1 )

If aN = {ax/x ∈ N} and bN ∩ cN = dN Where b, c ∈ N are relatively prime then

d = bc

c = bd

b = cd

a = bd

View Solution

Answer :

aN = {ax/x ∈ N}

bN = {bx/x ∈ N} = the set of positive integers multiple of b

cN = {cx/x ∈ N} = the set of positive integers multiple of c.

∴ bN ∩ cN = the set of positive integers multiple of bc.

∴ bN ∩ cN = bcN(∵ b and c are prime)

given – bN ∩ cN = dN

hence d = bc.

2 )

Two finite sets have m and n element respectively. The total number of subsets of first set is 112 more than the total number of sub sets of the second set. The value of m and n respectively are

5, 2

4, 7

7, 4

2, 5

View Solution

Answer :

No. of subset of first set – No of subset of second set = 112

∴ 2^m – 2ⁿ = 112

∴ 2ⁿ(2^m–n – 1) = 112

∴ 2ⁿ(2^m–n – 1) = 2⁴ × 7

∴ comparing n = 4

2^m–n – 1 = 7 i.e. 2^m–n = 8

hence

m – n = 3

∴ m = n + 3 = 7

Hence

m, n = 7, 4.

3 )

A survey shows that 70% of the Indian like mango where 82% like apple. If x% of Indian like both mango and apples then

x = 52

52 ≤ x ≤ 70

x = 70

70 ≤ x ≤ 82

View Solution

Answer :

n(A) = 70

n(B) = 82

x% like both mango and apple hence n(A ∩ B) = x

we know

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

∴ n(A ∩ B) = 70 + 82 – n(A ∪ B) = 152 – n(A ∪ B)

but n(A ∪ B) ≤ 100

hence

n(A ∩ B) ≥ 52 (1)

also n(A ∩ B) ≤ 70 (2)

Hence

52 ≤ n(A ∩ B) ≤ 70 from (1) & (2)

∴ 52 ≤ x ≤ 70.

4 )

If X ∪ {3, 4} = {1, 2, 3, 4, 5, 6} the which of the following is true

Smallest set X = {1, 2, 5, 6}

Smallest set X= {1, 2, 3, 5, 6}

Smallest set X= {1, 2, 3, 4}

Greatest set X= {1, 2, 3, 4}

View Solution

Answer :

X ∪ {3, 4} = {1, 2, 3, 4, 5, 6}

Then the Smallest set of X will be X ∪ {3, 4} – {3, 4} i.e.

{1, 2, 3, 4, 5, 6} – {3, 4} = {1, 2, 5, 6}.

5 )

If two sets A and B are having 43 elements in common, then the number of elements common to each of the sets A × B and B × A is

2^43

43^2

43^43

43^17

View Solution

Answer :

given: n(A ∩ B) = 43

No. of element common to both sets (A × B) and (B × A) is given by:

n[(A × B) ∩ (B × A)] = n(A ∩ B) × n(B ∩ A)

= 43 × 43 = 43².

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