Discrete Mathematics Answers

Question 16

Consider the following quantified statement:

∀x ∈ Z [(x2 ≥ 0) ∨ (x2 + 2x – 8＞0)].

Which one of the alternatives provides a true statement regarding the given statement or its 

negation?

1. The negation ∃x ∈ Z [(x2 < 0) ∨ (x2 + 2x – 8 ≤ 0)] is not true.

2. x = – 3 would be a counterexample to prove that the negation is not true.

3. x = – 6 would be a counterexample to prove that the statement is not true.

4. The negation ∃x ∈ Z [(x2 < 0) ∧ (x2 + 2x – 8 ≤ 0)] is true.

Let A = {□, ◊, ☼, ⌂} and let # be a binary operation from A  A to A presented by the 

following table:

# □ ◊ ☼ ⌂

□ □ ◊ ☼ ⌂

◊ ◊ □ ◊ □

☼ ☼ ◊ ☼ ⌂

⌂ ⌂ □ ⌂ ⌂

Answer questions 10 and 11 by referring to the table for #.

Question 10

Which one of the following statements pertaining to the binary operation # is TRUE?

1. ☼ is the identity element for #.

2. # is symmetric (commutative).

3. # is associative.

4. [(⌂ # ◊) # ☼] = [⌂ # (◊ # ☼)]

Question 11

# can be written in list notation. Which one of the following ordered pairs is an element of the list 

notation set representing #?

1. ((□, ◊), ⌂)

2. ((⌂, ☼), ◊)

3. ((☼, ◊), ◊)

4. ((⌂, ◊), ◊)

Answer questions 4 to 7 by using the given functions g and f. 

Hint: Drawing graphs of f and g before answering the questions, may assist you. Keep in mind 

that g  Z+  Q and f  Z+  Z+. Please note that graphs will not be asked for in the exam.

Question 6

Which one of the following alternatives represents the image of x under g ￮ f (ie g ￮ f(x)))?

1. 20x2 + 8x – 12

2. 80x2 + 4 x –

3. 20x2 + 8x + 3

4. 80x2 + 4 x – 3

Question 7

Which one of the following statements regarding the function g is TRUE? 

(Remember, g  Z+  Q.)

1. g can be presented as a straight line graph.

2. g is injective.

3. g is surjective.

4. g is bijective.

Question 3

Let G and L be relations on A = {1, 2, 3, 4} with

G = {(1, 2), (2, 3), (4, 3)} and L = {(2, 2), (1, 3), (3, 4)}.

Which one of the following alternatives represents the relation L ￮ G = G; L?

1. {(2, 3), (3, 3)}

2. {(1, 2), (2, 4), (4, 4)}

3. {(1, 2), (2, 1), (3, 3), (4, 4)}

4. {(2, 4), (4, 4)}

Suppose U = {1, 2, 3, 4, 5, a, b, c} is a universal set with the subset A = {a, b, c, 1, 2, 3, 4}.

Answer questions 1 and 2 by using the given sets U and A.

Question 1

Which one of the following relations on A is NOT functional?

1. {(1, 3), (b, 3), (1, 4), (b, 2), (c, 2)}

2. {(a, c), (b, c), (c, b), (1, 3), (2, 3), (3, a)}

3. {(a, a), (c, c), (2, 2), (3, 3), (4, 4)}

4. {(a, c), (b, c), (1, 3), (3, 3)}

Question 2

Which one of the following alternatives represents a surjective function from U to A?

1. {(1, 4), (2, b), (3, 3), (4, 3), (5, a), (a, c), (b, 1), (c, b)}

2. {(a, 1), (b, 2), (c, a), (1, 4), (2, b), (3, 3), (4, c)}

3. {(1, a), (2, c), (3, b), (4, 1), (a, c), (b, 2), (c, 3)}

4. {(1, a), (2, b), (3, 4), (4, 3), (5, c), (a, a), (b, 1), (c, 2)}

What is 10001000₂ - 0001101100₂

a)          Freddie has 6 toys cars and 3 toy buses, all different. 

i)              Freddie arranges these 9 toys in a line. Find the number of possible arrangements 

·           if there is a car at each end of the line and no buses are next to each other.

a)          Freddie has 6 toys cars and 3 toy buses, all different. 

i)              Freddie arranges these 9 toys in a line. Find the number of possible arrangements 

if the buses are all next to each other.

A fair six-sided dice is thrown and the scores are noted.

Event X: The total of the two scores is 4.

Even Y: The first score is 2 or 5.

a)              Construct the table by showing the sample spaces.                                          

a)              Write each statement in symbolic form using p, q and r.

If I study, then I will not fail mathematics.

If I do not play basketball, then I will study.

But I failed mathematics.

Then, test the validity of the following argument by using the truth table.