Search & Filtering
Find out which of the following functions from R to R are (i) One-to-one, (ii) Onto, (iii) One-to-one correspondence.
(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
Draw graphs of the following functions.
(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) = – 3x+4
(e)
f: R—>R defined by f(x) = Floor(x)
(f)
f: R—>R defined by f(x) = Ceiling(x)
(g)
f: R—>R defined by f(x) = x^2
(h)
f: R—>R defined by f(x) = x^3
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.
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.
Question 15
Let p, q and r be simple declarative statements. Which alternative provides the truth values for
the biconditional ‘’ of the compound statement provided in the given table?
Hint: Determine the truth values of p → r, q ∨ r, (p → r) (q ∨ r), q → p, ¬(q → p) and
¬(q → p) r in separate columns before determining the truth values of
[(p → r) (q ∨ r)] [¬(q → p) r].
p q r [(p → r) (q ∨ r)] [¬(q → p) r]
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
1.
T F
T T F
F T F
T F
T T F
F F T
F
T
F
T
T
F
F
T
2.
3.
4.
Question 14
Consider the truth table for the connective ‘’ with two simple declarative statements p and q.
p q p q
T T T
T F F
F T F
F F T
(a) Which one of the given alternatives represents ‘’ as a binary operation on the set of truth
values {T, F}? (b) Does the binary operation ‘’ have an identity element? For each alternative,
please look at (a) and (b).
1.
(a)
(b) The binary operation ‘’ does not have an identity element.
2.
(a)
T F
T F T
F T F
T F
T T F
F F F
(b) The binary operation ‘’ has an identity element.
3.
(a)
(b) The binary operation ‘’ does not have an identity element.
4.
(a)
(b) The binary operation ‘’ has an identity element.
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.
