Discrete Mathematics Answers

Is (p>q)>[(p>q)>q] a tautology? Why or why not?

Prove or disprove ¬(¬p -> q) =¬(p Ú q) is correct.

Show that (p → r) ∨ (q → r) and (p ∧ q) → r are logically equivalent. 

a)     (p ↔ q) ⊕ (¬p ↔¬r)

b)     (p → q) ∧ (¬p → r)

What are the differences between relations and functions?

Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one corre￾spondence.

(a)

f: R—>R defined by f(x) = x

(b)

f: R—>R defined by f(x) = |x|

(c)

f: R—>R defined by f(x) = x + 1

(d)

f: R—>R defined by f(x) = x^2

(e)

f: R—>R defined by f(x) = x^3

(f)

f: R—>R defined by f(x) = x – x^2

(g)

f: R—>R defined by f(x) = Floor(x)

(h)

f: R—>R defined by f(x) = Ceiling(x)

(i)

f: R—>R defined by f(x) = – 3x+4

(j)

f: R—>R defined by f(x)= – 3x^2 +7

Question 24

Consider the statement 

If n is a multiple of 3, then 2n + 2 is not a multiple of 3.

The converse of the given statement is:

 If n is not a multiple of 3, then 2n + 2 is a multiple of 3.

1. True

2. False

Question 25

Consider the following statement, for all x  Z:

 If x + 1 is even, then 3x2

- 4 is odd. 

The correct way to start a direct proof to determine if the statement is true is as follows:

 Assume x is even, then x = 2k for some k  Z,

 then 3x2 – 4

 ie 3(2k)2

- 4

 ie ………..

1. True

2. False

Question 22

Consider the following statement:

∀x  Z, [(2x + 4 > 0)  (4 - x

2 ≤ 0)]

The negation of the above statement is:

¬[∀x  Z, [(2x + 4 > 0)  (4 - x

2 ≤ 0)]]

≡ ∃x  Z, ¬[(2x + 4 > 0)  (4 - x

2 ≤ 0)]

≡ ∃x  Z, [¬(2x + 4 > 0) ∧ ¬(4 - x

2 ≤ 0)]

≡ ∃x  Z, [(2x + 4 ≤ 0) ∧ (4 - x

2 > 0)]

1. True

2. False

Question 23

Consider the statement

If n is even, then 4n2

- 3 is odd.

The contrapositive of the given statement is:

If 4n2

- 3 is odd, then n is even.

1. True

2. False

Question 17

Consider the following proposition:

For any predicates P(x) and Q(x) over a domain D, the negation of the statement 

∃x ∈ D, P(x) ∧ Q(x) 

is the statement

∀x ∈ D, P(x) → ¬Q(x).

We can use this truth to write the negation of the following statement:

“There exist integers a and d such that a and d are negative and a/d = 1 + d/a.”

Which one of the alternatives provides the negation of this statement?

1. There exist integers a and d such that a and d are positive and a/d = 1 + d/a.

2. For all integers a and d, if a and d are positive then a/d  1 + d/a.

3. For all integers a and d, if a and d are negative then a/d  1 + d/a.

4. For all integers a and d, a and d are positive and a/d  1 + d/a.