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Total questions: 127
Worksheet time: 1hrs 4mins
Let p be a proposition. The statement “It is not the case that p” is denoted by
p
¬p
p → q
p ∨ q
p ∧ q
Let p and q be propositions. The proposition that is true when both p and q are true and is false otherwise is denoted by
p ∧ q
p ∨ q
p → q
p ↔ q
p ∨ ¬q
Let p and q be propositions. The proposition that is false when p and q are both false and is true otherwise is denoted by
p ∨ q
p ∧ q
p → q
p ↔ q
¬p ∨ q
Let p and q be propositions. The proposition that is true when exactly one of p and q is true and is false otherwise is denoted by
p ∨ q
p ∧ q
p → q
p ↔ q
p ⊕ q
Let p and q be propositions. The proposition that is false when p is true and q is false and is true otherwise is denoted by
p → q
p ∨ q
p ∧ q
p ↔ q
¬(p ∧ q)
Let p and q be propositions. The proposition that is true when p and q have the same truth values and is false otherwise is denoted by
p ∨ q
p → q
p ∧ q
p ↔ q
p ⊕ q
Which of the following sentences is a proposition?
3 + 2 = 6
Can you help me?
Take this pencil.
x + 2 = 6
Why should you study discrete mathematics?
Find the converse of p → q.
q → p
¬p → ¬q
p → ¬q
¬q → ¬p
p ↔ q
Find the contrapositive of p → q.
¬q → ¬p
q → p
¬p → q
p ↔ q
¬p ↔ ¬q
Find the bitwise OR of the bit strings 1011 0010 and 0110 0110.
1111 0110
0010 0011
0011 0011
1010 1001
0111 1100
Find the bitwise AND of the bit strings 1010 1010 and 1001 1001.
1000 1000
1011 1011
0011 0011
1100 1100
0101 0101
Find the bitwise XOR of the bit strings 0111 0101 and 1101 0101.
1010 0000
1111 0111
0101 0100
0110 1010
1101 0101
Construct the truth table for the proposition (p ↔ q) → ¬(p ∨ q).
T T F F
T F T F
F T T F
F F T T
T F F T
Construct the truth table for the proposition (r → p) ↔ (q ∧ p).
T T T T
F F F F
T F T F
F T F T
T T F F
Let p, q and r be the propositions “You get an A on the final exam”, “You do every exercise in this book” and “You get an A in this class” respectively. Write the proposition “Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class” using p, q and r and logical connectives.
r ∧ q → p
q ∧ p → r
r → (p ∧ q)
p ∨ q → r
¬p ∧ ¬q → r
Let p and q be the propositions “You get an A on the final exam” and “You get an A in this class” respectively. Write the proposition “To get an A in this class, it is necessary for you to get an A on the final” using p, q and logical connectives.
q → p
p → q
p ↔ q
¬p → q
¬q → p
Evaluate the expression (0110 ∧ 0101) ⊕ (1100 ∧ 1010).
1100
1101
0010
1111
0101
Find the implication that is false.
If 2 + 3 = 5, then pigs can fly.
If 2 + 3 = 6, then God exists.
If 2 + 3 = 4, then 3 + 3 = 5.
If pigs can fly, then 1 + 3 = 5.
If 2 + 3 = 5, then 1 + 3 = 4.
Let p and q be the propositions “It is below freezing” and “It is snowing” respectively. Express the proposition (¬p ∨ q) ∧ (p → ¬q) as an English sentence.
It is either below freezing or it is snowing, but it is not snowing if it is below freezing.
It is below freezing or snowing, but it is not snowing only if it is below freezing.
It is either below freezing or snowing, and it is not below freezing if it is snowing.
It is below freezing but not snowing.
If it is below freezing, then it is not snowing.
Let p and q be the propositions “You miss the final examination” and “You pass the course” respectively. Express the proposition (p ↔ ¬q) as an English sentence.
If you don’t miss the final examination then you pass the course, and conversely.
That you don’t miss the final examination is necessary for passing the course.
Missing the final examination is sufficient for passing the course by you.
You don’t miss the final examination or you pass the course.
If you don’t pass the course, you miss the final examination.
A compound proposition is a tautology if
it is always true, no matter what the truth values of the propositions that occur in it.
it is always false, no matter what the truth values of the propositions that occur in it.
Find the proposition that is a tautology.
it is always true, no matter what the truth values of the propositions that occur in it.
it is always false, no matter what the truth values of the propositions that occur in it.
it is always true whenever each of the propositions that occur in it is true.
it is only false when each of the propositions that occur in it is false.
it is always true whenever all the propositions that occur in it have the same truth values.
Which of the following logical equivalences is a distributive law?
(r ∧ p) ∨ (q ∧ p) ↔ p ∨ (r ∧ q)
(r ∨ q) ∧ (p ∨ r) ↔ p ∨ (q ∧ r)
(p ∨ q) ↔ (q ∨ p)
(r ∧ q) ∧ (p ∨ r) ↔ p ∧ (q ∧ r)
Find the proposition that is logically equivalent to ¬(p ∨ q).
¬p ∧ ¬q
¬p ∨ ¬q
Which implication is logically equivalent to the implication ¬(s → r)?
¬(r → s)
s ∧ ¬r
¬s → r
¬(s ∧ r)
A proposition is a contingency if
it is neither a tautology nor a contradiction
it is both a tautology and a contradiction
it is not a contradiction
it is not a tautology
it is not logically equivalent to any tautology
Find a compound proposition involving the propositions p, q and r that is true when p and q are false and r is true, but is false otherwise.
¬p ∧ ¬q ∧ r
p ∨ q ∨ r
¬(p ∧ q) ∧ r
p ∧ q ∧ ¬r
Find a compound proposition involving the propositions p, q and r that is false when p is false and q and r are true, but is true otherwise.
¬p ∨ (q ∧ r)
p ∧ q ∧ ¬r
¬(p ∧ q) ∨ r
p ∨ (q ∧ r)
Find a compound proposition involving the propositions p, q and r that is true when p and q are true and r is false, but is false otherwise.
p ∧ q ∧ ¬r
¬(p ∧ q) ∨ r
p ∨ (q ∧ ¬r)
¬p ∨ ¬q ∨ r
Let P(x) be the statement 'x spends less than three hours every weekday in class', where the universe of discourse for x is the set of students. Express the proposition '¬∃x P(x)' in English.
There is a student who spends no less than three hours every weekday in class.
Every student doesn’t spend more than three hours every weekday in class.
There is a student who doesn’t spend more than three hours every weekday in class.
There is a student who spends less than two hours every weekday in class.
Every student spends no more than six hours every weekday in class.
Let P(x) be the statement 'x can speak Kazakh' and let Q(x) be the statement 'x knows the computer language Delphi', where the universe of discourse for x is the set of all students at your university. Express the sentence 'There is a student at your university who can speak Kazakh but who doesn’t know Delphi' in terms of P(x), Q(x), quantifiers and logical connectives.
∃x (P(x) ∧ ¬Q(x))
∃x (P(x) ∨ ¬Q(x))
∀x (P(x) → ¬Q(x))
∀x (¬P(x) ∨ Q(x))
Let S(x, y) be the statement 'x + 3y = 3x – y', where the universe of discourse for both variables is the set of integers. Which of the following statements is true?
∀x ∃y S(x, y)
∀x ∀y ¬S(x, y)
∃y ∀x S(y, x)
∃x ∀y S(x, y)
Rewrite the statement '¬∃y ∀x P(x, y)' so that negations appear only within predicates.
¬∀x ∃y P(x, y)
∀x ∀y P(x, y)
∀x ∃y ¬P(x, y)
∃y ∀x ¬P(x, y)
Rewrite the statement '¬∃y ∀x (S(x, y) ∨ ¬P(y))' so that negations appear only within predicates.
∀y ∃x (S(x, y) ∧ P(y))
∃y ∀x (S(x, y) ∧ P(y))
∀y ∀x (S(x, y) ∨ P(y))
∀y ∃x (S(x, y) ∨ ¬P(y))
Rewrite the statement '¬∃x ∀y (R(x, y) ∧ P(x))' so that negations appear only within predicates.
∀x ∃y (R(x, y) ∨ ¬P(x))
∀x ∀y (R(x, y) ∧ P(x))
∃x ∀y (¬R(x, y) ∨ ¬P(x))
∀x ∃y (¬R(x, y) ∧ P(x))
Rewrite the statement '¬∃z ∀y ∀x (S(z, x) ∧ P(z, y))' so that negations appear only within predicates.
∀z ∃y ∃x (S(z, x) ∨ P(z, y))
∀z ∀y ∀x (S(z, x) ∧ P(z, y))
∃z ∀y ∀x (¬S(z, x) ∨ ¬P(z, y))
∀z ∃y ∀x (¬S(z, x) ∧ P(z, y))
Which of the following statements is true if the universe of discourse for all variables is the set of all integers?
1 - 2 < m + n
1 >= n
2 = n
3 < m + n
2 <= m + n or m > n
Which of the following statements is true if the universe of discourse of each variable is the set of real numbers?
2 = x
2 = y
x + y ≠ y + x
3 = x
Find the power set of {1, 2}.
{∅, {1}, {2}, {1, 2}}
{∅, {1, 2}}
{{0}, {1}, {1, 2}}
{{0}, {1}}
{∅, {2}}
List the members of the set {x | x is a negative integer greater than (– 8)}.
{– 7, – 6, – 5, – 4, – 3, – 2, –1}
{– 8, –7, – 6, – 5, – 4, – 3, – 2, –1}
{–9, –10, –11, ...}
{0, 1, 2, 3, 4, 5, 6, 7}
{9, 10, 11, 12, ...}
List the members of the set {x | x is an integer such that 5 = x}.
∅
{5}
{–2, –1, 0, 1, 2}
{– 5; 5}
Find the power set of {0, 1}.
{∅, {0}, {1}, {0, 1}}
{∅, {0, 1}}
{{0}, {1}, {0, 1}}
{{0}, {1}}
Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8} and C = {2, 4}. Which of the following statements is true?
C ⊆ B ∩ A
B ⊆ C ∩ A
C ⊆ B ∪ A
A ⊆ C ∩ B
Let A = {a, b, c, d, e, f}, B = {b, d, f, g, h, s} and C = {m, n, o, p, r, d, f}. Find B ∩ C ∪ A.
{b, d, f, g, h, s}
{b, m, n, o, p, r, d, f}
{a, b, c, d, e, f}
{m, n, o, p, r, d, f}
Let A = {0, 1, 2}, B = {y, z} and C = {b, c}. Find C × B × A.
{(b, y, b), (b, y, c), (b, z, b), (b, z, c), (c, y, b), (c, y, c), (c, z, b), (c, z, c)}
{(0, y, b), (1, y, b), (2, y, b), (0, z, b), (1, z, c), (2, z, c), (0, y, c), (1, y, c), (2, y, c)}
{(b, y),(b, z), (c, y), (c, z)}
{(b, 0), (b, 1), (b, 2), (c, 0), (c, 1), (c, 2), (0, y), (1, y), (2, y)}
Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {0, 1, 3, 4, 7, 8}, B = {0, 2, 4, 6, 7, 8, 9}. Find B ∪ A.
{5}
∅
{0, 1, 2, 4, 6, 7, 8, 9}
{0, 1, 2, 3, 4, 6, 7, 8, 9}
Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {0, 1, 3, 4, 7, 8}, B = {0, 2, 4, 6, 7, 8, 9}. Find A - B.
{0, 1, 3, 4, 5, 7, 8}
{2, 6, 9}
{1, 3}
{5}
∅
A set A is a proper subset of a set B if
every element of A is also an element of B and A ≠ B
every element of A is also an element of B
there is an element of B that is not an element of A
there is an element of A that is also an element of B
Let A be a set. The power set of A is
the set of all subsets of A
the set of all proper subsets of A
{∅, A}
the set of all finite subsets of A
Let A and B be sets. The Cartesian product of A and B is
the set of all ordered pairs (a, b) where a ∈ A and b ∈ B
the set of all ordered pairs (b, a) where a ∈ A and b ∈ B
the set of all two-element sets {a, b} where a ∈ A and b ∈ B
the set of all ordered n-tuples where a ∈ A ∪ B for all i = 1, 2, ..., n.
Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {1, 3, 4, 5, 8}, B = {2, 4, 5, 7, 8}. Find the complement of A.
{2, 6, 7}
{1, 3, 6}
{4, 5, 8}
{6}
{1, 2, 3, 4, 5, 7, 8}
Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {1, 3, 4, 5, 8}, B = {2, 4, 5, 7, 8}. Find the complement of B.
{1, 3, 6}
{2, 6, 7}
{4, 5, 8}
{6}
{1, 2, 3, 4, 5, 7, 8}
Let A = {1, 2, 3, 4, 5} and A = {2, 3, 4}. Find the union of A.
{3, 4, 5, 6, 7}
{3}
{5, 6, 7}
{1, 2, 3, 4, 5}
{3, 4, 5}
Let A = {1, 2, 3, 4, 5} and A = {2, 3, 4}. Find the intersection of A.
{7, 8, 9,...}
{4, 5, 6, ...}
{1, 2, 3, ...}
{4, 5, 6, 7}
{1, 2, 3, 4}
Let f be a function from A to B. Then the codomain of f is
the set B
the set A
{b ∈ B | there is an element a ∈ A such that f(a) = b}
{a ∈ A | there is an element b ∈ B such that f(a) = b}
{b ∈ B | there is no a ∈ A with f(a) = b}
What is the Cartesian product of A = {a, b} and B = {1, 2}?
{(a, 1), (b, 1), (a, 2), (b, 2)}
{(1, a), (1, b), (2, a), (b, b)}
{(1, 1), (2, 2), (a, a), (b, b)}
{(1, a), (a, a), (1, b), (b, b)}
Let A = {a, b, c, d, e, g, h} and B = {0, 1, 3, 4, 5} with f(a) = 3, f(b) = 2, f(c) = 4, f(d) = 0, f(e) = 5, f(g) = 1 and f(h) = 3. Find the image of S = {c, d, e, g}.
(a)
Let A = {a, b, c, d, e, g, h} and B = {0, 1, 3, 4, 5} with f(a) = 3, f(b) = 2, f(c) = 4, f(d) = 0, f(e) = 5, f(g) = 1 and f(h) = 3. Find the image of S = {c, d, e, g}.
{0, 1, 4, 5}
{4, 0, 3, 1}
{0, 1, 2, 3, 4, 5}
{4, 0, 5, 1, 3}
{1, 2, 3, 4}
Which of the following is true for all sets S and T?
S ∩ T = ∅
T ∪ S ∈ T
T - S = ∅
T ∪ S = ∅
T ∩ S ≠ ∅
Which of the following is true for all sets S and T?
T ∪ S = T
T ∩ S = S
T ∪ S ∪ T = S
T ∪ S = T - S
T ∩ S = ∅
Let S = {a}. Which of the following is not an element of P(S) (the power set of S)?
{∅, {a}}
∅
{a}
{a, {a}}
{{a}}
Let S = {1}. Which of the following is an element of P(P(S)) (the power set of the power set of S)?
{∅, {1}}
{∅, 1}
{1}
{{∅}, 1}
{1, {1}}
What is the Cartesian product of A = {1, 2} and B = {a, b}?
{(1, a), (1, b), (2, a), (2, b)}
{(1, a), (1, b), (2, a), (b, b)}
{(1, 1), (2, 2), (a, a), (b, b)}
{(1, a), (a, a), (1, b), (b, b)}
{(1, 1), (a, a), (2, a), (1, b)}
Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {0, 1, 3, 4, 7, 8}, B = {0, 2, 4, 6, 7, 8, 9}. Find A ∩ B.
{1, 2, 3, 5, 6, 9}
∅
{0, 1, 2, 4, 6, 7, 8, 9}
{0, 4, 7, 8}
{4, 7, 8}
Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} be the universal set, and let A = {0, 1, 3, 4, 7, 8}, B = {0, 2, 4, 6, 7, 8, 9}. Find A ∩ B.
{5}
∅
{0, 1, 2, 4, 6, 7, 8, 9}
{0, 4, 7, 8}
{1, 2, 3, 5, 6, 9}
A function f: A → B is said to be injective (one-to-one) ...
iff f(x) = f(y) implies that x = y for all x and y in the domain of f.
iff for every element b ∈ B there is an element a ∈ A with f(a) = b.
if f(x) < f(y) whenever x < y and x and y are in the domain of f.
if f(x) > f(y) whenever x < y and x and y are in the domain of f.
iff for every element a ∈ A there is an element b ∈ B with f(a) = b.
A function f: A → B is said to be surjective (onto) ...
iff for every element b ∈ B there is an element a ∈ A with f(a) = b.
iff for every element a ∈ A there is an element b ∈ B with f(a) = b.
if f(x) < f(y) whenever x < y and x and y are in the domain of f.
if f(x) > f(y) whenever x < y and x and y are in the domain of f.
iff f(x) = f(y) implies that x = y for all x and y in the domain of f.
The function f is a one-to-one correspondence, or a bijection, if
it is both one-to-one and onto.
it is either one-to-one or onto.
it is one-to-one but not onto.
it is onto but not one-to-one.
the codomain and the range of f coincide.
Let f and g be the functions from the set of integers to the set of integers defined by f(x) = 3x – 4 and g(x) = 4x – 3. Find the composition of g and f.
12x – 19
12x – 13
7x – 7
7x – 12
x + 1
Find ⌈0.1, 1⌉.
2
1
1.01
0
1.02
Find ⌈-2, 3⌉.
-2
-3
-2, 3
0
-1
Find ⌈3/1 + 1/3⌉.
2
1
1/3
2/3
0
Which of the following functions from {a, b, c, d} to itself is one-to-one?
f(a) = d, f(b) = c, f(c) = b, f(d) = a
f(a) = c, f(b) = d, f(c) = b, f(d) = b
f(a) = a, f(b) = b, f(c) = d, f(d) = d
f(a) = a, f(b) = a, f(c) = a, f(d) = a
f(a) = d, f(b) = a, f(c) = a, f(d) = d
Which of the following functions from R to R is a bijection?
f(x) = 5x - 6
f(x) = 9x^2 - 2
f(x) = 1/(4x - 2)
f(x) = 1/(6 + x)
f(x) = 10x + 5
Let S = {-3, 0, 3, 7}. Find f(S) if f(x) = 2x + 3.
{1, 4, 17}
{17; 3; 3; 3}
{0, 3, 17}
{-2, 0, 3, 16}
{-3, 0, 3, 7}
Find g(f(x)) if f(x) = 2x + 3 and g(x) = 3x - 5 are functions from R to R.
2(3x - 5) + 3
7x + 3
1(3x - 5) + 3
3(2x + 3) - 5
3(2x + 3) + 5
Find ⌊3/4 - 3/2⌋.
0
-2/3
2
1
3
Find ⌊4/3 - 2/3⌋.
-2/3
4
1
2
3
Suppose f: Z → R has the rule f(n)=2n-3. Which of the following is the range of f?
the set of odd integers
the set of even integers
the real numbers
the set of natural numbers {0, 1, 2, 3, ...}
the set of integers {..., -2, -1, 0, 1, 2, 3, ...}
Suppose f and g are functions from R to R with f(x) = 2x and g(x) = 3 - 2x. Find (g ∘ f)(5).
-1
1
0
2
5
Suppose f and g are functions from R to R with f(x) = 2x + 3 and g(x) = 3 - 2x. Find (g ∘ f)(5).
0
1
-1
3
5
Suppose N → N has the rule f(n) = { 5, if n ≥ 8; 1, if n < 8 }. Which of the following statements is true?
f is not a function because f(6) is equal to both 36 and 7
f is a function
f is not a function because f(3)=9 and f(8)=9
none of this
f is not a function because there is no natural number n such that f(n)=2
Find functions R to R f and R to R g such that g f has the rule (2.8 + 2) = x.
f(x) = x + 2.8, g(x) = x
f(x) = 2.8, g(x) = x + 2
g(x) = 2 + x, f(x) = x
g(x) = 2.8, f(x) = x + 2
Let f: A to B where B = {0, 1, 4, 9} and f is defined by the rule f(x) = x^2. For which set A is f a correctly defined function from A to B and one-to-one?
{1, 2, 3}
{-2, -1, 0, 1, 2}
{-3, -2, 0, 1, 2}
{0, 1, 2}
Suppose the function A to R f is defined by the rule f(n) = 3 - n. On which domain is the function 1-1 and onto?
{3, 4, 6}
{2, 3, 4, 5}
{3, 5}
{2, 3, 4}
Suppose the function Z to A f is defined by the rule f(n) = 3 - n. On which domain A is the function 1-1?
{1, -1, 3, 4}
{-1, 0, 1, 3}
{-2, -1, 0, 1}
{1, 2, 3, 4}
Find 99.
1
2
1, 98
0
1, 99
Suppose R to R f where f(x) = 2 - x. Find {1} where x = f(x).
{3, 5}
{2, 0}
{3, 2}
none of these
Suppose R to R f where f(x) = 3 - x. Find {2} where x = f(x).
{3, 7}
{3, 0}
{3, 2}
none of these
Suppose R to A f has the rule f(x) = 3 - x. Which of the following could be the domain A?
{-1, 0, 1}
{-1, 0}
{-1, -1}
none of these
Which is a correct statement about the function f(x) = x^2 from the set of integers Z to the set of integers?
f is one-to-one, but not onto
f is onto, but not one-to-one
f is one-to-one and onto
f is neither one-to-one nor onto
Which statement about the function f(x) = x + 1 from the set of real numbers to the set of integers is correct?
f is onto, but not one-to-one
f is one-to-one, but not onto
f is one-to-one and onto
f is neither one-to-one nor onto
The function B to {0, 1, 2, 3} where f(n) = 2 - n is an onto function if the codomain B is which of the following?
{..., -2, -1, 0, 1, 2, 3}
{3, 2, 1, 0}
R (the set of all real numbers)
{..., -2, -7, -5, -3, -1, -1}
Suppose the function A to {0, 1, 2, 3} is defined by the rule f(n) = 3 - n. On which domain A is the function onto the codomain {0, 1, 2, 3}?
{6, 4, 3, 1}
{5, 3, 1}
{5, 3, 2, 1}
{6, 5, 3, 2}
The function R to A f where f(x) = 2 - x is one-to-one if the domain A is which of these sets?
{-π, π}
{x | x < 2}
Z (the set of all integers)
{-2, 1, 3}
There are 20 mathematics majors and 40 computer science majors at a college. How many ways are there to pick one representative who is either a mathematics major or a computer science major?
60
800
20
40
80
There are 20 mathematics majors and 40 computer science majors at a college. How many ways are there to pick two representatives, so that one is a mathematics major and another is a computer science major?
800
60
20
40
80
How many strings of five English letters that start with B are there if letters can be repeated?
4 * 26
22
23
24
25
If set A has 3 elements then number of elements in A × A × A is?
27
3
9
12
6
Let A = {a, b, c, d, e, f}. How many 4-combinations of A are there?
15
360
60
30
120
Find C(8, 5).
56
70
35
5
840
Find the coefficient of y^57 in (12)(y + x).
792
21
72
99
664
Find the coefficient of 4y^2 in (6)(5x - 3y).
3 * 5
15
3 * 3
2 * 4
-15
How many different strings can be made from the letters in STATISTICS, using all the letters?
50400
100800
10!
C(10, 3)
720
A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants and broccoli croissants. How many ways are there to choose a dozen croissants?
6188
12376
924
210
3003
How many ways are there to choose a dozen croissants?
6188
12376
924
210
3003
How many ways are there to choose eight coins from a piggy bank containing 100 identical pennies and 80 identical nickels?
9
C(100, 8) + C(80, 8)
P(100, 80)
36
!8!80!100
Find P(8, 5).
6720
336
56
28
72
Find C(8, 4).
70
1680
35
140
840
A dormitory has students – 12 sophomores, 8 juniors, and 20 seniors. Which of the following is equal to the number of ways to form a committee of five students from the same class?
C(12, 5)+ C(8, 5)+ C(20, 5)
C(40, 5)/3
P(12, 5)+P(8, 5)+P(20, 5)
P(20, 5) P(8, 5) P(12, 5)
C(20, 5) C(8, 5) C(12, 5)
A class consists of 12 women and 10 men. How many ways are there to form a committee of size six if the committee has more women than men?
C(10, 6) C(12, 0) + C(10, 5) C(12, 1) + C(10, 4) C(12, 2)
C(10, 4) C(12, 2)
C(10, 6) C(12, 1) + C(10, 5) C(12, 2) + C(10, 4) C(12, 3)
C(10, 5) C(12, 1)
C(10, 4) C(12, 2)
How many different license plates are available if the license plate pattern consist of three letters that cannot be repeated followed by three digits that can be repeated (Assume that all letters are uppercase and the digits are 0, 1, ..., 9.)?
3
10
24
25
26
A class consists of 12 women and 10 men. How many ways are there to form a committee of size six if the committee has equal numbers of women and men?
C(10, 3) C(12, 3)
C(12, 3) + C(10, 3)
C(22, 6) - C(12, 6) - C(10, 6)
C(22, 6)
C(22, 3)
If you were to use a Caeser cipher with a key of 3 shifts, how would you encrypt the word: “CIPHER”?
FLSKHU
FMKSHU
EKRJGT
BLOGDQ
DJQIFS
“HELLO” has been encrypted with Caesar cipher to “KHOOR”. The key used was?
3
2
1
4
5
What word encrypted using a Caesar Cypher and the key 3 would be: “WRGDB”?
TODAY
MAKER
CODED
LESSON
TERMS
What will be the plain text corresponding to cipher text “PROTO” if Vigenere cipher is used with keyword as “HELLO”.
INDIA
WORLD
AMERICA
TABLE
ITEMS
What will be the ciphered text if the string “INDIA” is given as input to the code of Vigenere cipher with keyword as “HELLO”.
PROTO
PARTO
RAPYO
RAPTO
TOPRO
The quotient and remainder when -1 is divided by 3 is?
-1 and 2
1 and 2
-1 and -2
-1 and 1
2 and 1
Which positive integer less than 21 are relatively prime to 21?
19
18
21
24
22
What is the quotient q and remainder r when -11 divided by 3?
q=-4 and r=1
q=1 and r=4
q=11 and r=0
q=-3 and r=4
q=1 and r=-3
What is the quotient q and remainder r when 547 divided by 4?
q=136 and r=3
q=3 and r=136
q=136 and r=0
q=3 and r=130
q=136 and r=-3
What is the quotient q and remainder r when -111 divided by 11?
q=-11 and r=10
q=11 and r=10
q=111 and r=0
q=-10 and r=11
q=16 and r=-3
Represent the relation R = {(1, 1), (1, 2), (2, 1), (3, 2)} on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order).
[0, 1, 0], [0, 0, 1], [0, 1, 1]
[1, 0, 0], [1, 1, 1], [0, 0, 0]
[0, 1, 1], [0, 0, 1], [1, 1, 0]
[1, 0, 1], [0, 1, 0], [1, 0, 1]
[0, 0, 0], [1, 0, 1], [1, 1, 1]
List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3} where (a, b) ∈ R if and only if a + b = 3.
{(0, 3), (1, 2), (2, 1), (3, 0)}
{(0, 3), (1, 2), (2, 1), (3, 0), (4, –1)}
{(0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}
{(0, 0), (1, 1), (2, 2), (3, 3)}
{(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 4)}
Let R1 = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3), (3, 4)} and R2 = {(1, 2), (2, 1), (2, 4), (3, 1), (3, 2), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find R1 ∩ R2.
{(1, 2), (3, 4)}
{(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4)}
{(1, 1), (2, 2), (2, 3), (3, 3)}
{(1, 1), (2, 1), (2, 4), (3, 1), (3, 2)}
{(1, 1), (1, 2), (2, 2), (3, 3), (3, 4)}
Let R = {(a, b), (b, c), (c, a), (d, b)} and S = {(a, a), (b, b), (c, c), (d, a)} be relations on A = {a, b, c, d}. Find R ∘ S.
{(a, b), (b, c), (c, a), (d, b)}
{(a, a), (b, b), (c, c), (d, a)}
{(a, a), (b, a), (c, a), (c, b), (d, b), (d, c)}
{(a, a), (a, b), (b, b), (b, c), (c, a), (c, c), (d, a), (d, b)}
{(b, a), (c, b), (a, c), (b, d), (a, d)}
