Relations and Functions — Practice

Question 1

Check whether the relation R = {(1,1),(2,2),(3,3),(1,2)} on the set {1,2,3} is reflexive.

Accepted Answer

For reflexivity, (1,1), (2,2), and (3,3) must all be in R. All three are present. Answer: R is reflexive.

Question 2

Is f: R -> R, f(x) = 5x - 3 one-one?

Accepted Answer

f(x1)=f(x2) implies 5x1-3=5x2-3 implies x1=x2. Answer: Yes, f is one-one.

Question 3

Show that R = {(a,b) : a-b is divisible by 3} on the set of integers is an equivalence relation.

Accepted Answer

Reflexive: a-a=0, divisible by 3. Symmetric: if 3 divides a-b, then 3 divides -(a-b)=b-a. Transitive: if 3 divides a-b and b-c, then 3 divides (a-b)+(b-c)=a-c. Answer: All three hold, so R is an equivalence relation.

Question 4

Show that f: R -> R, f(x) = x^3 is onto.

Accepted Answer

For any y in R, choose x = y^(1/3), which is a real number since cube roots of all reals exist. Then f(x) = (y^(1/3))^3 = y. Answer: Every y in R has a preimage, so f is onto.

Question 5

If f(x) = x+1 and g(x) = 2x, find (g o f)(x) and (f o g)(x). Are they equal?

Accepted Answer

(g o f)(x) = g(f(x)) = g(x+1) = 2(x+1) = 2x+2. (f o g)(x) = f(g(x)) = f(2x) = 2x+1. Answer: 2x+2 is not equal to 2x+1, so gof is not equal to fog.

Question 6

Show that f: R -> R defined by f(x) = 7x - 4 is invertible, and find f^{-1}.

Accepted Answer

One-one: 7x1-4=7x2-4 implies x1=x2. Onto: for any y, x=(y+4)/7 gives f(x)=y. So f is bijective, hence invertible. Solving y=7x-4 for x: x=(y+4)/7. Answer: f^{-1}(y) = (y+4)/7.

Question 7

For the equivalence relation R = {(a,b) : a and b leave the same remainder when divided by 4} on the set {0,1,...,11}, find the equivalence class of 2.

Accepted Answer

Elements leaving remainder 2 when divided by 4, within {0,...,11}: 2, 6, 10. Answer: [2] = {2, 6, 10}.

Question 8

f: {1,2,3,4} -> {1,2,3,4} is a one-one function. Without further information, is f necessarily onto?

Accepted Answer

Both domain and co-domain are the same finite set of size 4. Answer: Yes — for a function from a finite set to itself, one-one always implies onto.

Practice — Relations and Functions

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