Discrete Mathematics Answers

Show that the square of an even number is an even number using a direct proof.

Simplify

1. p∧(p∧q)

2. ~ (~p∨q)

3. ~ (p⇒~q)

let p(n) 1^3+2^3+....+n^3=(n(n+1)/2)2 for positive integer n

Is function f defined by f = {(1, 2), (3, 4), (5, 6), (8, 6), (10, -1)}, a one to one function?

In a computer science department, a student club can be formed with either 10 members from first year or 8 members from second year or 6 from third year or 4 from final year. What is the minimum no. of students we have to choose randomly from department to ensure that a student club is formed?

Determine the functions f: R -> R are onto, f(x) = |x| + x

Let A = {1, 2, 3, 4} and B = {0, 3, 6, 8, 12, 15}. Consider a rule f (x) = x² - 1, x∈A, then show that f is a mapping from A to B.

The following sets have been defined using the | notation. Re-write them by listing some of the elements.

i. {p | p is a capital city, p is in Europe}

ii. {z | 3z = n2, z and n are natural numbers}

Use a direct proof to show that the sum of two odd integers(k and l) is even.

1.      Evaluate each of these expressions.

a)  1 1000 ∧ (0 1011 ∨ 1 1011)

b) (0 1111 ∧ 1 0101) ∨ 0 1000

c)  (0 1010 ⊕ 1 1011) ⊕ 0 1000

d) (1 1011 ∨ 0 1010) ∧ (1 0001 ∨ 1 1011)