Discrete Math - Chapter 2
What is another name for a function?
Mapping/Transformation
What set does N represent
Natural numbers
What is the cardinality of a set?
Number of elements in a finite set
What is a sequence?
Ordered list of elements
What is the notation for a power set?
P(S)
What set does Z+ represent
Positive integers
What set does Z* represent?
Positive integers plus zero
What set does Q represent?
Rational numbers
What set does R represent?
Real numbers
What does A intersection B do?
Returns a set with only elements appearing in both sets
What does A complement B do?
Returns set with elements of A that did not appear in B
What does a floor function do?
Round down to nearest integer
What is a singleton set?
Set with only one element (could be a set containing only the empty set)
What is a summation?
Sum of the terms in a sequence
What is an integer valued function
The co-domain is the set of integers
What is a real valued function
The co-domain is the set of real numbers
What makes a set countable (other than clearly finite sets)?
The set has the same cardinality of positive integers
What is the power set of a set?
The set of all subsets
What does it mean if two sets are equal?
They have all of the same elements
What is a set
Unordered collection of objects
How is { a <= x < b } represented in set interval notation
[a, b)
How is { a <= x <= b } represented in set interval notation
[a, b]
For the function f(a) = b, where f: A -> B, what is the pre-image of b
a
What is the form of an arithmetic progression?
a, a + d, a + 2d, ..., a + nd
What is the form of a geometric progression?
a, ar, ar^2,...,ar^n
𝐴={a,𝑏} 𝐵={1,2,3} Select the the expression that is an element of 𝐴×𝐵×𝐵 . a. (𝑏,2,3) b. (𝑎,𝑎,1) c. (𝑏,22) d. (2,1,1)
a. (𝑏,2,3)
Select the value of ⌈−5.8⌉ . a. -5 b. -6 c. 5 d. 6
a. -5
A = {1, 2, {3, 4}, {5, 6, 7}} Select the correct value for |A|. a. 4 b. 5 c. 6 d. 7
a. 4
Select the law that establishes that the two sets below are equal. (𝐴∩𝐵)∪(𝐴∩𝐵)=𝐴∩𝐵 a. Idempotent law b. Identity law c. Absorption law d. Distributive law
a. Idempotent law
𝑓:{0,1}^3→{0,1}^3 f(x) is obtained by removing the second bit from x and placing the bit at the end of the string. For example, f(101) = 110. Select the correct description of the function f. a. One-to-one and onto b. One-to-one but not onto c. Onto but not one-to-one d. Neither one-to-one nor onto
a. One-to-one and onto
𝑓:𝐙→𝐙. 𝑓(𝑥)=𝑥+3 Select the correct description of the function f. a. One-to-one and onto b. One-to-one but not onto c. Onto but not one-to-one d. Neither one-to-one nor onto
a. One-to-one and onto
𝐴={x∈𝐙:𝑥is even} 𝐵={x∈𝐙:𝑥is a prime number} 𝐷={5,7,8,12,13,15} Select the set corresponding to 𝐷−(𝐴∪𝐵) . a. {15} b. {13,15} c. {8,12,15} d. {5,7,13,15}
a. {15}
𝐵={x∈𝐙:𝑥is a prime number} 𝐶={3,5,9,12,15,16} The universal set 𝑈 is the set of all integers. Select the set corresponding to 𝐵∩𝐶 . a. {3,5} b. {9,12,16} c. {3,5,9,15} d. {9,12,15,16}
a. {3,5}
𝐴={x∈𝐙:𝑥is even} 𝐶={3,5,9,12,15,16} 𝐷={5,7,8,12,13,15} The universal set 𝑈 is the set of all integers. Select the set corresponding to NOT[(𝐴∪𝐷)]∩𝐶 . a. {3,9} b. {8,12} c. {5,12,15} d. {5,7,13,15}
a. {3,9}
Select the function that has a well-defined inverse. a. 𝑓:𝐙→𝐙 𝑓(𝑥)=𝑥+4 b. 𝑓:𝐙→𝐙 𝑓(𝑥)=2𝑥−5 c. 𝑓:𝐙→𝐙+ 𝑓(𝑥)=|𝑥| d. 𝑓:𝐙→𝐙 𝑓(𝑥)=⌈𝑥/2⌉
a. 𝑓:𝐙→𝐙 𝑓(𝑥)=𝑥+4
What is a recurrence relation?
an equation that expresses a term of a sequence in terms of one or more of the previous terms
For the function f(a) = b, where f: A -> B, what is the image of a
b
Define strings s = 101 and t = 10. Select the string that is equal to st. a. 101101 b. 10110 c. 101 d. 10
b. 10110
Select the value of ⌊4.2⌋ . a. 0 b. 4 c. 4.2 d. 5
b. 4
The string x is equal to 1101. What is the length of 𝜆𝑥 ? a. 3 b. 4 c. 5 d. 6
b. 4
𝑓:𝐙+→𝐙+. 𝑓(𝑥)=𝑥+3 Select the correct description of the function f. a. One-to-one and onto b. One-to-one but not onto c. Onto but not one-to-one d. Neither one-to-one nor onto
b. One-to-one but not onto
A = {a, b, c, d} X = {1, 2, 3, 4} The function 𝑓:𝐴→𝑋 is defined by the arrow diagram below. Select the set of pairs that defines a function that is equal to f. a -> 2; b -> 3; c -> 4; d -> 2 a. f = {(a, 2), (b, 3), (d, 2)} b. f = {(a, 2), (b, 3), (c, 4), (d, 2)} c. f = {(a, 2), (b, 3), (c, 4), (d, 4)} d. f = {(a, 1), (b, 3), (c, 4), (d, 4)}
b. f = {(a, 2), (b, 3), (c, 4), (d, 2)}
Select the collection of sets that forms a partition of: {1, 2, 3, 4, 5, 6, 7, 8} a. {1, 2, 5, 7} {3, 4} {8} b. {1, 2, 5, 7} {3, 4, 6} {8} c. {0, 1, 2, 5, 7} {3, 4, 6, 8} d. {1, 2, 5, 7} {3, 4, 6, 8} {2, 4}
b. {1, 2, 5, 7} {3, 4, 6} {8}
𝐴={x∈𝐙:𝑥is even} 𝐶={3,5,9,12,15,16} 𝐷={5,7,8,12,13,15} Select the set corresponding to 𝐶−(𝐴⊕𝐷) . a. {3,9,16} b. {3,9,12} c. {3,5,9,15} d. {3,7,8,9,13,16}
b. {3,9,12}
𝐴={x∈𝐙:𝑥is a prime number} 𝐵={4,7,9,11,13,14} Select the set corresponding to 𝐴∩𝐵 . a. ∅ b. {7, 11, 13} c. {7, 9, 11, 13} d. {4, 7, 9, 11, 13, 14}
b. {7, 11, 13}
Select the collection of sets that forms a partition of 𝐙 . a. 𝐙+,𝐙− b. 𝐙+,𝐙−,{0} c. 𝐙,𝐙− d. 𝐙,𝐙−,{0}
b. 𝐙+,𝐙−,{0}
Select the set that is equivalent to 𝐶∪(𝐶∩𝐵) a. ∅ b. 𝐶 c. 𝐶∪𝐵 d. 𝐵∩𝐶
b. 𝐶
𝐴={x∈𝐙:𝑥is even} 𝐶={3,5,9,12,15,16} Select the true statement. a. 𝐶−𝐴={12,16} b. 𝐶−𝐴={3,5,9,15} c. 𝐶−𝐴={3,5,9,12,15} d. The set 𝐶−𝐴 is infinite.
b. 𝐶−𝐴={3,5,9,15}
What is the term for a function that is one-to-one and onto?
bijective
𝑓:{0,1}^4→{0,1}^4 f(x) is obtained by removing the second bit from x and placing the bit at the end of the string. For example, f(1011) = 1110. Select the correct value for 𝑓−1(0101) . a. 1010 b. 0101 c. 0110 d. 0011
c. 0110
Two functions, f and g, map real numbers to integers. The functions f and g are defined as: 𝑓(𝑥)=⌊𝑥+1/2⌋ 𝑔(𝑥)=⌈𝑥−1/2⌉ Select the value for x such that 𝑓(𝑥)≠𝑔(𝑥) . a. 0 b. 2 c. 2.5 d. 2.75
c. 2.5
𝑓:𝐙→𝐙. 𝑓(𝑥)=⌈𝑥/3⌉ Select the correct description of the function f. a. One-to-one and onto b. One-to-one but not onto c. Onto but not one-to-one d. Neither one-to-one nor onto
c. Onto but not one-to-one
A = {a, b, c, d} X = {1, 2, 3, 4} Select the definition for f that is a well-defined function. a. f = {(a, 2), (b, 3), (d, 1)} b. f = {(a, 2), (b, 3), (b, 3), (d, 1)} c. f = {(a, 2), (b, 3), (c, 3), (d, 1)} d. f = {(a, 2), (b, 3), (c, 3), (1, d)}
c. f = {(a, 2), (b, 3), (c, 3), (d, 1)}
A = {a, b, c, d} X = {1, 2, 3, 4} Each choice defines a function whose domain is A and whose target is X. Select the function that has a well-defined inverse. a. f = {(a, 3), (b, 4), (c, 3), (d, 4)} b. f = {(a, 3), (b, 3), (c, 3), (d, 3)} c. f = {(a, 3), (b, 4), (c, 2), (d, 1)} d. f = {(a, 3), (b, 4), (c, 2), (d, 4)}
c. f = {(a, 3), (b, 4), (c, 2), (d, 1)}
A = {a, b, c, d} X = {1, 2, 3, 4} The function 𝑓:𝐴→𝑋 is defined as f = {(a, 4), (b, 1), (c, 4), (d, 4)} Select the set corresponding to the range of f. a. {∅} b. {1} c. {1, 4} d. {1, 2, 3, 4}
c. {1, 4}
A = {1, 2, {3, 4}, {5, 6, 7}} Select the statement that is true. a. {3}∈𝐴 b. {3,4}⊆𝐴 c. {1,2}⊆𝐴 d. {1,2}∈𝐴
c. {1,2}⊆𝐴
For 𝑖∈𝑍+ , 𝐴𝑖 is defined to be the set of all integer multiples of 𝑖 . Select the set corresponding to (⋂[𝑖=2 to 4]𝐴𝑖) ∩ {x∈𝐙:1≤𝑥≤30} a. ∅ b. {24} c. {12, 24} d. {6, 12, 18, 24, 30}
c. {12, 24}
𝐴={x∈𝐙:𝑥is a prime number} 𝐵={4,7,9,11,13,14} 𝐶={x∈𝐙:3≤𝑥≤10} Select the set corresponding to (𝐴∪𝐵)∩𝐶 . a. {3, 5, 7} b. {3, 4, 7, 9} c. {3, 4, 5, 7, 9} d. {3, 4, 5, 7, 9, 11, 13}
c. {3, 4, 5, 7, 9}
𝐶={3,5,9,12,15,16} 𝐷={5,7,8,12,13,15} Select the set corresponding to 𝐶⊕𝐷 . a. {3,9,16} b. {5,12,15} c. {3,7,8,9,13,16} d. {3,5,7,8,9,12,13,15,16}
c. {3,7,8,9,13,16}
Select the set that is equal to: 3,5,7,9,11,13 a. {x∈𝐙:3<𝑥<14} b. {x∈𝐑:3≤𝑥<14} c. {x∈𝐙:𝑥is odd and3≤𝑥≤14} d. {x∈𝐙:𝑥is prime and3≤𝑥<14}
c. {x∈𝐙:𝑥is odd and3≤𝑥≤14}
What are integers?
-3,-2,-1,0,1,2,3
What makes a function one-to-one?
Each element of the co-domain is only mapped to an element of the domain once (not assigned to multiple domain elements)
Describe what is necessary for set A to be a subset of set B
Every element of A must also be an element of B
What makes a function onto?
Every element of the co-domain is mapped to an element of the domain
What set does Z represent
Integers
Select the collection of sets that forms a partition of 𝐑 . a. {𝑥∈𝐑:𝑥<2} {𝑥∈𝐑:2<𝑥<4} {𝑥∈𝐑:4≤𝑥} b. {𝑥∈𝐑:𝑥<4} {𝑥∈𝐑:2≤𝑥≤4} {𝑥∈𝐑:2<𝑥} c. {𝑥∈𝐑:𝑥<2} {𝑥∈𝐑:2≤𝑥<4} {𝑥∈𝐑:4≤𝑥} d. {𝑥∈𝐑:𝑥≤2} {𝑥∈𝐑:2≤𝑥<4} {𝑥∈𝐑:4≤𝑥}
c. {𝑥∈𝐑:𝑥<2} {𝑥∈𝐑:2≤𝑥<4} {𝑥∈𝐑:4≤𝑥}
A donut store sells packages of 12 donuts. The store has made x donuts. How many complete packages does the store have for sale? a. ⌊12𝑥⌋ b. ⌈12𝑥⌉ c. ⌊𝑥/12⌋ d. ⌈𝑥/12⌉
c. ⌊𝑥/12⌋
Select the statement that is false. a. 𝐙⊂𝐑 b. 𝐙+⊂𝐍 c. 𝐙⊂𝐑+ d. 𝐙⊆𝐑
c. 𝐙⊂𝐑+
Select the function that does not have a well-defined inverse. a. 𝑓:𝐙→𝐙 𝑓(𝑥)=⌈𝑥+2⌉ b. 𝑓:𝐑→𝐑 𝑓(𝑥)=−2𝑥+5 c. 𝑓:𝐑→𝐙 𝑓(𝑥)=⌈𝑥⌉ d. 𝑓:𝐑→𝐑 𝑓(𝑥)=3𝑥+4
c. 𝑓:𝐑→𝐙 𝑓(𝑥)=⌈𝑥⌉
A and B are finite sets. The function 𝑓:𝐴→𝐵 is a bijection. Select the true statement. a. 𝑓∘𝑓^−1=𝑓 b. 𝑓∘𝑓^−1=𝐼(𝐴) c. 𝑓∘𝑓^−1=𝐼(𝐵) d. f may not have a well-defined inverse.
c. 𝑓∘𝑓^−1=𝐼(𝐵)
In an arithmetic progression, what is the name of the 'd' term?
common difference
In a geometric progression, what is the name of the 'r' term?
common ratio
Select the law that establishes that the two sets below are equal. 𝐴∩NOT[(𝐵∪𝐶)]=𝐴∩(NOT[𝐵]∩NOT[𝐶]) a. Distributive law b. Associative law c. Absorption law d. De Morgan's law
d. De Morgan's law
A = {1, 2, 3, 4}. Select the statement that is false. a. ∅∈𝑃(𝐴) b. ∅⊆𝑃(𝐴) c. {2,3}∈𝑃(𝐴) d. {2,3}⊆𝑃(𝐴)
d. {2,3}⊆𝑃(𝐴)
Use the definition below to select the statement that is false. 𝐴={x∈𝐙:𝑥is even and4<𝑥<17} a. 4∉𝐴 b. 6∈𝐴 c. 17∉𝐴 d. |𝐴|=7
d. |𝐴|=7
Use the definitions below to select the statement that is true. 𝐴={x∈𝐙: 𝑥 is even} 𝐵={x∈𝐙: −4<𝑥<17} a. A is finite. b. 𝐵⊆𝐴 c. 𝐴⊂𝐴 d. ∅⊂𝐵
d. ∅⊂𝐵
Select the set that is equivalent to (𝐵∩𝐶)∪∅ a. ∅ b. 𝐵 c. 𝐶 d. 𝐵∩𝐶
d. 𝐵∩𝐶
Rewrite the composition (f o g)(a) to more clearly define what is taking place
f(g(a))
In a geometric progression, what is the name of the 'a' term?
initial term
In an arithmetic progression, what is the name of the 'a' term?
initial term
What is the other word for a one-to-one function?
injective
What common number sets are countable?
integers, natural numbers, rational numbers
How is an empty string denoted?
lambda
What is required for a function to have an inverse?
must be a bijection
What common number sets are uncountable?
real numbers
What is the other word for an onto function?
surjective
What is P({0,1,2})
{empty set, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}
What are natural numbers?
Counting numbers: 1, 2, 3, ......
What does A union B do?
Combines all elements of A and all elements of B into a set
For the function f(a) = b, where f: A -> B, what is the domain
A
What is the difference between a subset and a proper subset?
A proper subset is one directional, meaning the sets are not equal
What are rational numbers?
All numbers which can be expressed as a ratio of two integers. (All integers and fractions.) (-2, 1, 1/4, 1/2)
What are irrational numbers?
An Irrational Number is a real number that cannot be written as a simple fraction. Ex: Square Root of 2, 3, 24, 10, 41
For the function f(a) = b, where f: A -> B, what is the co-domain
B
What is the name of the solution to A x B
Cartesian product
What set does C represent?
Complex numbers
