Relation and Function

Discuss the subjectivity of the following functions.

Give examples of relation which are

(i) Neither reflexive nor symmetric nor transitive.                

(ii) symmetric and reflexive but not transitive.

(iii) Reflexive and transitive but not symmetric.

Check the injectivity and subjectivity of the following functions:

Let Z be the set of all integers. A relation R is defined on Z by x R y to mean x – y is divisible by 5. Show that R is an equivalence relation on Z.

Find whether the following functions are one-one:

Show that the relation R in R defined as R ={(a, b); a ≤ b} is reflexive and transitive but not symmetric

Prove that the relation R defined on set N of natural numbers by x Ry ⟺2x² – 3xy + y² = 0 i.e., by 

Determine which of the following binary operations on the set N are associative and which are commutative.

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L₁, L₂): L₁ is parallel to L₂}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Show that the Modulus function f : R → R given by f(x) = |x|, is neither one one nor onto, where  |x| is x, if x is positive or 0 and |x| is –x, if x is negative.

Show that the relation R in the set A of point in a plane given by R = {(P, Q)}: distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Prove that the greatest integer function f : R → R given by f(x) = [x], is neither one one nor onto, where [x]denotes the greatest integer less than or equal to x.

Let A = R – {3} and B = R – {1}. Consider the function f: a → B defined by

f one-one and onto?  Justify your answer.  

Let A, B are two sets, show that f : A × B → B × A. Show that f(a, b) = (b, a) is bijective function.

Give examples of relation which are

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric

Let f : N → N be defined by

for all n ∈ N, state whether the function f is bijective. Justify your answer.

Let f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {y ∈ N: y = 4x + 3 for some x∈ N}. Show that f is invertible. Find the inverse.

Show that *: R × R ⊠⊠ R given by a * b ⊠⊠ a + 2b is not associative.

If ‘*’ is defined on the set R of real numbers by a*b = 3ab/7, find the identity element in R for the binary operation ‘*’.

Give an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive. 

(ii) Transitive but neither reflexive nor symmetric. 

(iii) Reflexive and symmetric but not transitive. 

(iv) Reflexive and transitive but not symmetric. 

(v) Symmetric and transitive but not reflexive.