Discrete Mathematics Answers

Q : 5). If f is function from A to B and g is function B to C and both f and g are onto.
Show that go f is also onto. Is go f one-to-one if both f and g are one-to-one.


Q : 6) Let f, g and h: R → R be defined by (R is the set of real numbers)
f(x) = x + 2, g(x) = (1 + x2)
-1, h(x) = 3.
Compute f -1g(x) and hf (g f -1) (hf(x)).

Q : 1) Determine whether each function is one-to-one. The domain of each
function is the set of all real numbers. If the function is not one-to-one,
prove it. Also, determine whether f is onto the set of all real numbers. If f is
not onto, prove it.
a) f(x) = 6x – 9
b) f(x) = 2x3 - 4

Q : 2) Let A = {1, 2, 3}, B = {p, q} and C = {a, b}. Let f: A → B is f = {(1, p), (2, p), (3,
a)} and g: B → C is given by {(p, b), (q, b)}. Find go f and show it pictorially.

Q:1. Determine whether each set is a function from X = {1,2,3,4} to Y = {a,b,c,d}.
If it is a function, find its domain and range, draw its arrow diagram, and
determine if it is one-to-one, onto or both.
a) {(1,a,),(2,a),(3,c),(4,b)}
b) {(1,c),(2,a),(3,b),(4,c),(2,d)}
c) {(1,d),(2,d),(4,a)}

Q:2. List all possible functions from A to B, A = {a, b, c}, B = {0, 1}. Also indicate in
each case whether the function is one-to-one, is onto and one-to-one-onto.

Let R = {(2, 2), (2, 4), (3, 3), (3, 6), (3, 12), (4,2), (6, 3)}
where if a is a factor b}

(i) Draw the digraph for the relation P.
(ii) Find the domain and range of the relation.

Describe two situations that can be modelled with a tree . Describe the graph model that you will use

Describe two situations that can be modelled with a Tree. Describe the graph model that you will use.

Describe two situations that can be modelled with a Graph. Describe the graph model that you will use

Prove that for all integers n with 1≤n≤10,n^2 −n+11 is prime.

Prove that for all positive integers n
3^n +7^n −2 is divisible by 8

Let X = {0, 1, 2}. Consider the power set P(X). Recall that P(X) = {A such that A ⊆ X}.
Determine if each statement is true or false. (a) X ∈ P(X).
(b) ∅∈P(X)
(c) X ⊆ P(X).
(d) {0,2}∈P(X) (e) {0,2,1} ∈ X (f) {0}⊆X
(g) {{0}} ⊆ P(X)