Q: If n(X)= ^m C_6, then the value of m is

$S=\ 1,2,3,4,5,6\ R: S S$ Number of elements in $R=6$ and for each $(a, b) R ;|a-b| 2$ $X $ set of all relation $R: S S$ If $a=1, b=3,4,5,6 (4)$ $a=2, b=4,5,6 (3)$ $a=3, b=1,5,6 (3)$ $a=4, b=1,2,6 (3)$ $a=5, b=1,2,3 (3)$ $a=6, b=1,2,3,4 (4)$ Total number of ordered pairs $(a, b)$ s.t. $|a-b| 2 =20$ $ n(X)=$ number of elements in $ X = ^ 20 C_6$ $ m=20$

Question 1

Let S = \ 1, 2, 3, , 10\ . Consider the set X = \ R : R is an equivalence relation on the set S such that R has exactly 42 elements\ . Then the number of elements in X is _____________.

Accepted Answer

An equivalence relation on a set S corresponds to a partition of S into disjoint equivalence classes. Let the sizes of these equivalence classes be a_1, a_2, , a_k. The number of elements in the equivalence relation R is given by the sum of the squares of the sizes of its equivalence classes: _ i=1 ^k a_i^2 = 42 Since the total number of elements in S is 10, we also have: _ i=1 ^k a_i = 10 We need to find all possible partitions of 10 such that the sum of their squares is 42. Let's check the pos

Question 2

Let the set of all relations R on the set \ a, b, c, d, e, f\ , such that R is reflexive and symmetric, and R contains exactly 10 elements, be denoted by S. Then the number of elements in S is ______

Accepted Answer

For relation to be reflexive all the diagonal elements must be taken and out of remaining 30 elements there are 15 pairs and we need 2 pairs such that R contains exactly 10 elements and is both reflexive and symmetric. number of ways = ^ 15 C _2=105

Question 3

If n(X)= ^m C_6, then the value of m is

Accepted Answer

$S=\ 1,2,3,4,5,6\ R: S S$ Number of elements in $R=6$ and for each $(a, b) R ;|a-b| 2$ $X $ set of all relation $R: S S$ If $a=1, b=3,4,5,6 (4)$ $a=2, b=4,5,6 (3)$ $a=3, b=1,5,6 (3)$ $a=4, b=1,2,6 (3)$ $a=5, b=1,2,3 (3)$ $a=6, b=1,2,3,4 (4)$ Total number of ordered pairs $(a, b)$ s.t. $|a-b| 2 =20$ $ n(X)=$ number of elements in $ X = ^ 20 C_6$ $ m=20$

Question 4

If the value of n(Y)+n(Z) is k^2, then |k| is Let $S=\ 1,2,3,4,5,6\ $ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly

Accepted Answer

given |a-b| 2 so if i.e. Total elements in X is ^ 20 C _6 Now for n ( Y ), range of R has exactly one element i.e. second elements must be constant in R and since R must have 6 element so it is not possible to satisfy both condition so n ( Y )=0. aligned & for n(z) \ & 1 3,4,5,6 \ & 2 4,5,6 \ & 3 1,5,6 \ & 4 1,2,6 \ & 5 1,2,3 \ & 6 1,2,3,4 \ & aligned no. of relation that are function will be = ^4 C _1 ^3 C _1 ^3 C _1 ^3 C _1 ^3 C _1 ^4 C _1 aligned & =(4 3 3)^2=k^2 \ & i.e. k =36 aligne

Question 5

In a study about a pandemic, data of  900 persons was collected. It was found that 190  persons had symptom of fever, 220  persons had symptom of cough, 220  persons had symptom of

Accepted Answer

We denote the set of people having symptoms of fever, cough and breathing problem by  F , C  and  B  respectively Given that  n F = 190 , &#160; n B = 220  and  n C = 220 Also given,  n F &#8746; C = 330 , &#160; n B &#8746; C = 350 ,   n F &#8746; B = 340  and  n F &#8745; C &#8745; B = 30 So  n F &#8745; C = n F + n C - n F &#8746; C = 80 Similarly,  n F &#8745; B = n F + n B - n F &#8746; B = 70  and  n C &#8745; B = n

Sets and Relations — Previous Year Questions & Solutions