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
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