Question 1 :
Let X = {a, b, c, d} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
Solution :
(i) reflexive
X = {a, b, c, d} and R = {(a, a), (b, b), (a, c)
The relations R has the elements (a, a) and (b, b), but the set X contains two more elements c and d. In order to make the given set as reflexive, we have to include the elements (c, c) and (d, d).
(ii) symmetric
"R is said to be symmetric if a is related to b implies that b is related to a"
In the given relation we have (a, c), so we have to add the element (c, a) to make the relation as symmetric.
(iii) transitive
R is said to be transitive, if “a is related to b and b is related to c” implies that a is related to c.
We have (a, a) (a, c) and (a, c). Hence it is transitive.
(iv) equivalence
A relation on R is said to be an equivalence relation if it is reflexive, symmetric and transitive.
To make the given relation as equivalence, we have to add the elements (c, c) (d, d) and (c, a).
Question 2 :
Let A = {a, b, c} and R = {(a, a), (b, b), (a, c)}. Write down the minimum number of ordered pairs to be included to R to make it
(i) reflexive (ii) symmetric (iii) transitive (iv) equivalence
Solution :
(i) reflexive
A = {a, b, c} and R = {(a, a), (b, b), (a, c)}
The relations R has the elements (a, a) and (b, b), but there is another element c. In order to make the given set as reflexive, we have to include the element (c, c).
(ii) symmetric
"R is said to be symmetric if a is related to b implies that b is related to a"
In the given relation we have (a, c), so we have to add the element (c, a) to make the relation as symmetric.
(iii) transitive
R is said to be transitive, if “a is related to b and b is related to c” implies that a is related to c.
From the given relation itself, we have transitivity.
(iv) equivalence
A relation on R is said to be an equivalence relation if it is reflexive, symmetric and transitive.
To make the given relation as equivalence, we have to add the elements (c, c) and (c, a).
Question 3 :
Let P be the set of all triangles in a plane and R be the relation defined on P as aRb if a is similar to b. Prove that R is an equivalence relation
Solution :
Reflexive :
Every triangle is similar to itself. Hence it is reflexive.
Symmetric :
If the triangle a is similar to b, then the triangle b is also similar to a. Hence it is symmetric.
Transitive :
If the triangle a is similar to b and a is similar to c, then a will also be similar to c. Hence it is transitive.
Since the given relation satisfies reflexivity, symmetricity and transitivity, it is known as equivalence relation
Question 4 :
f(x) = {(2, 2) (3, 3) (4, 4) (5, 5) (6, 6)} be a relation of set A = {2, 3, 4, 5, 6}, then it is a
a) Reflexive and transitive b) Reflexive and symmetric
c) Reflexive only d) An equivalence relation.
Solution :
If a is a element belongs to the set A, if a is related to a. Then it is in reflexive relation. It should be true for all the elements containing in the set A.
That is all elements in set A should be associated with the same element.
Here in the set A, we have the elements 2, 3, 4, 5 and 6. By considering the relation, they all are associated with the same elements. So, it is reflexive. Hence option c is correct.
Question 5 :
If a is related to b if and only if the difference in a and b is an even integer. This relation is
a) Symmetric, reflexive but not transitive
b) Symmetric, transitive but not reflexive
c) Transitive, reflexive but not symmetric
d) Equivalence relation.
Solution :
Check for reflexivity :
a) a relation is reflexive if every element has a relation with itself.
b) In this question, the relation exists only if the difference between the elements is an even integer.
c) Take for example, the number 2. Now, for this relation to be a reflexive relation, this element 2 would have to have a relation with itself.
d) 2 - 2 = 0, which is even integer.
e) Therefore, any element can have a relation with itself and hence this is reflexive relation.
Check for Symmetric :
a) A relation is symmetric if (a, b) ∈ R ==> (b, a) ∈ R.
b) Take two integers, 2 and 6
c) Here 2 - 6 = -4, which is even integer.
d) Also 6 - 2 = 4, which is an even integer.
e) Therefore (2, 6) ∈ R and (6, 2) ∈ R
f) Therefore this is a symmetric relation.
Check for Transitivity :
a) A relation is transitive if (a, b) ∈ R and (b, c) ∈ R ==> (a, c) ∈ R
b) Take the values of a, b and c to be 2, 6, and 10 respectively.
c) Now a = 2, b = 6 and c = 10
d) Clearly (a, b) ∈ R as 2 - 6 = -4, which is an even integer.
e) Also (b, c) ∈ R as 6 - 10 = -4, which is an even integer.
f) Also (a, c) ∈ R as 2 - 10 = -8, which is an even integer.
g) Therefore, this relation is transitive relation.
Since this relation is Reflexive, Symmetric and transitive relation, it is am equivalence relation.
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