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:**

Thus, R is not reflexive

Therefore, R is symmetric

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}

And hence, R is reflexive.

Again, since x an y have same number of pages

Thus, R is symmetric

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:**

Thus, R is reflexive

Therefore, R is symmetric.

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

is an equivalence relation. Find the set of all elements related to 1 in each case.

**Solution:**

= {(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)}

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}

Thus, R is reflexive.

Thus, R is symmetric

Thus, R is transitive.

Therefore, R is equivalence relation.

The set of elements related to 1 = {1}

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