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A relation R on set A = {1, 2, 3, 4, 5} is defined as R = {(x, y) : |x² – y²| < 8}. Check whether the relation R is reflexive, symmetric and transitive.
A function f is defined from R → R as f(x) = ax + b, such that f(1) = 1 and f(2) = 3. Find function f(x). Hence, check whether function f(x) is one-one and onto or not.
Show that a function f : R → R defined by f(x) = 2x/(1 + x²) is neither one-one nor onto. Further, find set A so that the given function f : R → A becomes an onto function.
A relation R is defined on N × N (where N is the set of natural numbers) as : (a, b) R (c, d) ⇔ a – c = b – d. Show that R is an equivalence relation.
A relation R on set A = {– 4, – 3, – 2, – 1, 0, 1, 2, 3, 4} be defined as R = {(x, y) : x + y is an integer divisible by 2}. Show that R is an equivalence relation. Also, write the equivalence class [2].
Assertion (A): The relation R = {(x, y) : (x + y) is a prime number and x, y ∈ N} is not a reflexive relation. Reason (R): The number '2n' is composite for all natural numbers n.
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