Q5. Let R be the relation on Z defined by R = {(a, b) : a, b ∈ Z, a − b is an integer}. Find the domain and range of R (NCERT Exercise 2.2, Q9).
The difference of any two integers is an integer, so every pair of integers is in R: R = Z × Z. Domain = range = Z.
Q6. Write the relation R = {(x, y) : x + 2y = 10; x, y ∈ N} in roster form, and give its domain and range.
x = 10 − 2y must be a natural number, so y = 1, 2, 3 or 4: R = {(8, 1), (6, 2), (4, 3), (2, 4)}; domain {2, 4, 6, 8}; range {1, 2, 3, 4}.
Q7. A set A has 3 elements. How many relations are there on A? How many of them contain every pair (a, a) with a ∈ A?
A × A has 9 pairs, so there are 29 = 512 relations on A. If the 3 pairs (a, a) must be in, only the other 6 pairs are free: 26 = 64. These are exactly the reflexive relations on A; a set with n elements has 2n² − n of them.
Q8. n(A) = 2 and n(B) = 3. How many relations from A to B are non-empty? How many contain one given pair?
There are 26 = 64 relations; leaving out ϕ gives 63 non-empty ones. A given pair is in exactly half of them: 25 = 32.
Q9. Show that R = {(a, b) : a, b ∈ Z and a − b is even} is reflexive, symmetric and transitive.
a − a = 0 is even, so (a, a) ∈ R. If a − b is even, so is b − a = −(a − b). If a − b and b − c are even, so is their sum a − c. So R is an equivalence relation, the same pattern as NCERT Example 19.