Class 12 Mathematics

Relations And Functions

NCERT Solution of Math Exercise 1.1 (Part - 4)

Question – 6: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.

Solution:

class 12 math Relations and functions NCERT Solution52 Exercise 1.1

Thus, R is not reflexive

class 12 math Relations and functions NCERT Solution53 Exercise 1.1

Therefore, R is symmetric

class 12 math Relations and functions NCERT Solution54 Exercise 1.1

Thus, R is not transitive.

Hence, R is symmetric but not reflexive or transitive.


Question – 7 – Show that the relation R in the set of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages}, is an equivalence relation.

Solution:

Given, by R = {(x, y): x and y have same number of pages}

class 12 math Relations and functions NCERT Solution55 Exercise 1.1

And hence, R is reflexive.

Again, since x an y have same number of pages

class 12 math Relations and functions NCERT Solution56 Exercise 1.1

Thus, R is symmetric

class 12 math Relations and functions NCERT Solution57 Exercise 1.1

Because number of pages in x and z is same.

Thus, R is transitive.

Hence, R is reflexive as well as symmetric and transitive.

Thus, R is an equivalence realtion.

Question – 8 – Show that the relation R in the set A = {1, 2, 3, 4, 5}, given by R = {(a, b): |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Solution:

class 12 math Relations and functions NCERT Solution58 Exercise 1.1

Thus, R is reflexive

class 12 math Relations and functions NCERT Solution59 Exercise 1.1

Therefore, R is symmetric.

class 12 math Relations and functions NCERT Solution60 Exercise 1.1

Thus, R is transitive.

Now, the elements of {1, 3, 5} are related to each other.

Because |1 – 3| = 2;

| 3 – 5 | = 1, and | 1- 5 | = 4

And all numbers are even numbers.

Similarly, elements of (2, 4) are related to each other.

Because, |2 – 4| = 2, which is even number.

But, no element of set, {1, 3, 5} is related to any element of {2, 4}

Because, | 1 – 2| = 1; |3 – 2| = 1; |5 – 2|= 3; |3 – 4| = 1 and | 5 – 4| =1, which are not even numbers.

Hence, no element of {1, 3, 5} is related to any element of {2, 4}



Question – 9 – Show that the relation R in the set class 12 math Relations and functions NCERT Solution61 Exercise 1.1
class 12 math Relations and functions NCERT Solution62 Exercise 1.1
is an equivalence relation. Find the set of all elements related to 1 in each case.

Solution:

class 12 math Relations and functions NCERT Solution63 Exercise 1.1

= {(0, 0), (0, 4), (0, 8), (0, 12), (1, 1), (1, 5), (1, 9), (2, 2), 2, 6), 2, 10), 3, 3), (3, 7), (3, 11), (4, 4), (4, 8), (4, 12), 5, 5), (5, 9), (6, 6), (6, 10), (7, 7), (7, 11), (8, 8), (8, 12), (9, 9), (10, 10), (11, 11), (12, 12)}

class 12 math Relations and functions NCERT Solution64 Exercise 1.1

Therefore, R is transitive.

Thus, R is reflexive, symmetric and transitive. Thus, R is an equivalence relation.

The set of elements related to 1 is equal to {1, 5, 9}

class 12 math Relations and functions NCERT Solution65 Exercise 1.1

Thus, R is reflexive.

class 12 math Relations and functions NCERT Solution66 Exercise 1.1

Thus, R is symmetric

class 12 math Relations and functions NCERT Solution67 Exercise 1.1

Thus, R is transitive.

Therefore, R is equivalence relation.

The set of elements related to 1 = {1}


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