Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.
Show that the relation R in the set N × N defined by (a, b) R (c, d) if a² + d² = b² + c² is an equivalence relation.
Let R be a relation defined on the set of natural numbers N as follow :
R = {(x, y) : and 2x + y = 24}
Find the domain and range of the relation R. Also,find if R is an equivalence relation or not.
Let A = {1, 2, 3, . ...., 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d =b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also obtain the equivalence class [(2, 5)].
Let f : X -> Y be a function. Define a relation R on X given be R = {(a, b) : f(a) = f(b)}. Show that R is an equivalence relation ?
If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.
Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.
Show that the relation R on the set Z of all integers defined by (x, y) ∈ is divisible by 3 is an equivalence relation.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by f(x) = Show that f is bijective. Also, find
(i) x, if = 4
(ii)
Let f : W → W be defined as show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.
Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by show that f is one-one and onto. Hence find
Show that the function f in A = defined as f(x) = is one-one and onto. Hence find
Consider f : given by f(x) = x² + 4 Show that f is invertible with the inverse of f given by = , where is the set of all nonnegative real numbers.
Let f : N → N be defined as
for all n ∈ N. State whether the function f is
bijective. Justify your answer.
Show that f : N → N, given by f(x) = |``x+1, if x is odd x- 1, if x is even is both one-one and onto.``|