Discrete Mathematics Answers

1. a) Determine whether the following proposition is logically equivalent or not using the truth table.

((p⟶q)v(q⟶p))⇔p⟷q

b) Construct a truth table to determine whether the following compound statement is a tautology, a contradiction or a contingency.

~(p∧r)⟶~(q∨r)

c) Use the laws of logic to establish the following logical expression.

~(p∨q) v (~p∧q)⟺~p

Determine whether the statement is logically equivalent using truth tables. [ (~P ∧ Q ) ⊕ P ] AND ( ~P V∧ Q)

Use Euclidean algorithm to determine the gcd (4076, 1024)

There are 6 people to be arranged in a line for a concert. How many

arrangements are possible?

Prove by PMI for 2^n+1>2n+1

Two functions f : R → R and g : R → R are defined by f(x) = 5x3 + 1 and g(x) = 2x − 3 for all x ∈ R.

Determine the inverse of (f -1 ◦ g) and (g ◦ f )(2) and ( f ◦ g)(2).

Solve the inequalities. Give your answer in interval notation, and indicate the answer

geometrically on the real-number line.

a. t + 6 ≤ 2 + 3t

b. 3(2 – 3x) > 4(1 – 4x)

Let the function f : R → R and g : R → R be defined by f(x) 2x + 3 and g(x) = -3x + 5.

a. Show that f is one-to-one and onto.

b. Show that g is one-to-one and onto.

c. Determine the composition function g o f

d. Determine the inverse functions f -1 and g -1 .

e. Determine the inverse function (g o f) -1 of g o f and the composite f -1 o g -1 .