Discrete structure Determine whether each of these functions is a bijection from

Question

Discrete structure Determine whether each of these functions is a bijection from R to R (R is the set of real numbers), and provide explanation (a) f(x) = 2x + 1 (b) f(x) = x^2 + 1 (c) f(x) = x^3 (d) f(x) = (x^2 + 1)/(x^2 + 2) Let R be the relation {(1, 2), (0, 3), (2, 3), (2, 4), (3, 1)}, and let S be the relation {(2, 1)(3, 1) (3, 2)(4, 2)} Find S compositefunction R. Demonstrate the relation R = {(1, 1), (2, 1), (3, 2), (4, 3)} represented in ordered pairs using (1) digraph, and (2) the matrix. Find the reflexive closure, symmetric closure, and transitive closure of above relation R.

Explanation / Answer

The answer is (a) and (c)

A bijective function between two sets states that each element of one set is paired with exactly one element of the other set.

Let's first see option b and d

b. f(-1) = f(1) = 2, so it clearly cannot be bijective
d. f(-1) = f(1) = 2/3, and so, it cannot be a bijective function

a. f(x) = 2*x+1 uniquely maps each each value of x with unique value of f(x)

f(-2) = -3
f(-3/2) = -2
f(-1) = -1
f(-1/2) = 0
f(0) = 1
f(1/2) = 2
f(1) = 3
f(2) = 5
....
.... so on, and it holds true for all real numbers R


c. f(x) = x3 uniquely maps each each value of x with unique value of f(x)

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