Unit 2 Flashcards
A blank is a collection of objects
set
The objects in a set are called blank
elements
The blank definition of a set is a list of the elements enclosed in curly braces with the individual elements separated by commas. The following definition of the set A uses roster notation:
A = { 2, 4, 6, 10 }
roster notation
The symbol blank is used to indicate that an element is in a set, as in 2 ∈ A.
∈
The symbol blank indicates that an element is not in a set, as in 5 ∉ A.
∉
The set with no elements is called the blank and is denoted by the symbol ∅
empty set
The empty set is sometimes referred to as the blank and can also be denoted by {}. Because the empty set has no elements, for any element a, a ∉ ∅ is true.
null set
A blank has a finite number of elements.
finite set
An blank has an infinite number of elements.
infinite set
The blank of a finite set A, denoted by |A|, is the number of elements in A.
cardinality
Two sets are blank if they have exactly the same elements
equal
The set of natural numbers: All integers greater than or equal to 0.
N
The set of all integers.
Z
The set of rational numbers: All real numbers that can be expressed as a/b, where a and b are integers and b ≠ 0.
Q
The set of real numbers
R
The superscript blank is used to indicate the positive elements of a particular set. For example, the set R+ is the set of all positive real numbers, and Z+ is the set of all positive integers.
+
The superscript blank is used to indicate the negative elements of a particular set. For example, the set R- is the set of all negative real numbers, and Z- is the set of all negative integers.
-
In blank, a set is defined by specifying that the set includes all elements in a larger set that also satisfy certain conditions. The notation would look like:
A = { x ∈ S : P(x) }
S is the larger set from which the elements in A are taken. P(x) is some condition for membership in A. The colon symbol “:” is read “such that”. The description for A above would read: “all x in S such that P(x)”.
set builder notation
The blank, usually denoted by the variable U, is a set that contains all elements mentioned in a particular context.
universal set
Sets are often represented pictorially with blank. A rectangle is used to denote the universal set U, and oval shapes are used to denote sets within U.
Venn diagrams
If every element in A is also an element of B, then A is a blank of B, denoted as A ⊆ B If there is an element of A that is not an element of B, then A is not a subset of B, denoted as A ⊈ B. If the universal set is U, then for every set A:
∅ ⊆ A ⊆ U
subset
Two sets are blank if and only if each is a subset of the other:
A = B if and only if A ⊆ B and B ⊆ A
equal
If A ⊆ B and there is an element of B that is not an element of A (i.e., A ≠ B), then A is a blank of B, denoted as A ⊂ B.
proper subset
The blank is a subset of every set.
empty set
Every set is a blank of itself.
subset
Set A is a subset of B, denoted blank if every element in A is also in B.
A⊆B
Set A is a proper subset of set B if it is a subset of B and is not equal to B. It is denoted blank
A⊂B
Sets can contain sets as blank.
elements
The cardinality of a set within a set is blank. That is, each set counts as a single element.
1
If S = the set of all sets that are not members of itself, then is S a member of itself?
If S is in S then it cannot be a member of S by definition of being in S.
The blank of a set A, denoted P(A), is the set of all subsets of A. For example, if A = { 1, 2, 3 }, then:
P(A) = { ∅, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }
power set
Since |A| = 3, the cardinality of the power set of blank
8
Let A be a finite set of cardinality n. Then the cardinality of the power set of blank
A is 2^n, or |P(A)|=2^n.
The power set of a set A is denoted blank
P(A)
The power set of any set A is the set of all blank, including the empty set and A itself.
subsets of A
The blank is an element of every power set.
empty set
The blank of the power set of a set of size n is 2^n
cardinality
Let A and B be sets. The intersection of A and B, denoted blank and read “A intersect B”, is the set of all elements that are elements of both A and B.
A ∩ B
A = { x ∈ Z: x is an integer multiple of 2 }
B = { x ∈ Z: x is an integer multiple of 3 }
A ∩ B = blank
{ x ∈ Z: x is an integer multiple of 6 }
The union of two sets, A and B, denoted blank and read “A union B”, is the set of all elements that are elements of A or B.
A ∪ B
A = { x: student x received an A on midterm 1 }
B = { x: student x received an A on midterm 2 }
Then
A ∪ B = blank
= { x: student x is eligible to skip the final exam }
{ x: student x received an A on midterm 1 or midterm 2 }
The blank of two sets: A∪B
union
The blank of two sets: A∩B
intersection
The union and intersection operations are blank. That is, A∪B=B∪A and A∩B=B∩A
commutative
Set operations can be blank to define even more sets.
combined
The expression A ∩ B ∩ C ∩ D is blank because the order in which intersection operations are applied does not matter.
well-defined
Similarly, the expression A ∪ B ∪ C ∪ D is also blank and defines the set consisting of those elements that are elements of at least one of the four sets: A, B, C, and D.
well-defined
The intersection and union operators can be combined in sequence: A∩(B∪C) or A∪(B∩C). Make sure to complete operations blank first.
inside parentheses
The blank between two sets A and B, denoted A - B, is the set of elements that are in A but not in B.
difference
The difference operation is not blank since it is not necessarily the case that A - B = B - A
commutative
The symmetric difference between two sets, A and B, denoted blank, is the set of elements that are a member of exactly one of A and B, but not both. An alternative definition of the symmetric difference operation is:
A ⊕ B = ( A - B ) ∪ ( B - A )
A ⊕ B
The symmetric difference is blank. A - B = B - A
commutative
The blank of a set A, denoted Ä, is the set of all elements in U that are not elements of A. An alternative definition of Ä is U - A. For example, let U = Z, and define:
A = { x ∈ Z: x is odd }
complement
x ∈ A ∩ B ↔ blank
(x ∈ A) ∧ (x ∈ B)
x ∈ A ∪ B ↔ blank
(x ∈ A) ∨ (x ∈ B)
x ∈ Å ↔ blank
¬(x ∈ A)
x ∈ ∅ ↔ blank
F
x ∈ U ↔ blank
T
A blank is an equation involving sets that is true regardless of the contents of the sets in the expression. The idea is similar to an equivalence in logic which holds regardless of the truth values of the individual variable in the expressions.
set identity
A ∪ A = A
Idempotent laws
A ∩ A = A
Idempotent laws
To say A union A or to say A intersect A is redundant. In both cases, we just have A.
Idempotent laws
A ∪ (A ∩ B) = A
Absorption laws
A ∩ (A ∪ B) = A
Absorption laws
A union any subset of A is just A.
Absorption laws
(A ∪ B) ∪ C = A ∪ (B ∪ C)
Associative laws
(A ∩ B) ∩ C = A ∩ (B ∩ C)
Associative laws
Only applies when the operators are all ∪ or ∩. Rearranging the parentheses in an expression will not change its value.
Associative laws
A ∪ B = B ∪ A
Commutative laws
Rearranging the operator within a single union or intersection will not change its truth value.
Commutative laws
A ∩ B = B ∩ A
Commutative laws
A ∩ Å = ∅
~U= ∅
Complement laws
A ∪ Å = U
∅ = U
Complement laws
It cannot be true that an element is in both A and the complement of A. Similarly, all elements in U are either in A or its complement.
Complement laws
A ∪ B = A ∩ B (all superscript–)
De Morgan’s laws
A ∩ B = A ∪ B (all superscript)
De Morgan’s laws
If the complement operation is distributed in a parenthesized expression with either the union or intersection operation, change the union operation to intersection operation, and vice versa.
De Morgan’s laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributive laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Distributive laws
This law is similar to the distributive law of multiplication over addition of the real numbers. For example, 5(x+y) = 5x + 5y
Distributive laws
A ∩ ∅ = ∅
Domination laws
A ∪ U = U
Domination laws
The intersection of a set with the empty set is always empty set. The union of a set A with the universal set is always the universal set.
Domination laws
The complement of the complement of a set is the set itself.
Double Complement law
A ∪ ∅ = A
Identity laws
A ∩ U = A
Identity laws
The union of A with itself can be reduced to just A. The intersection of A with itself can be reduced to just A
Identity laws
To blank a set identity, work with one side of the equation and manipulate it by using the identity laws until it matches the other side of the equation.
prove
An blank of items is written (x, y)
ordered pair
The first blank of the ordered pair (x, y) is x and the second entry is y.
entry
For two sets, A and B, the blank of A and B, denoted A x B, is the set of all ordered pairs in which the first entry is in A and the second entry is in B. That is:
A x B = { (a, b) : a ∈ A and b ∈ B }
Since the order of the elements in a pair is significant, A x B will not be the same as B x A, unless A = B. If A and B are finite sets, then |A x B| = |A|⋅|B|.
Cartesian product
A = { 1, 2, 3 }
B = { x, y }
(1, y) ∈ A x B
True or False
True
An ordered list of three items is called an blank and is denoted (x, y, z).
ordered triple
For n ≥ 4, an ordered list of n items is called an blank. For example, (w, x, y, z) is an ordered 4-tuple and (u, w, x, y, z) is an ordered 5-tuple.
ordered n-tuple
The blank of three sets contains ordered triples, and for n ≥ 4, the blank of n sets contains n-tuples. The blank of n sets, A1, A2, …, An is
A1 x A2 x … x An = { (a1, a2, … , an) : ai ∈ Ai for all i such that 1 ≤ i ≤ n }
For example, define A = {a, b}, B = {1, 2}, C = {x, y}, and D = {α, β}. Then the 4-tuples (a, 1, y, β) and (b, 1, x, α) are both examples of elements in the set A × B × C × D.
Cartesian product
The Cartesian product of a set A with itself can be denoted as A × A or blank More generally:
For example, if A = {0, 1}, then An is the set of all ordered n-tuples whose entries are bits (0 or 1). For n = 3:
{0, 1}^3 = { (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1) }
A^2.
The product A1×A2… ×An = {a1∈A1, a2∈A2, …, an∈An}. The elements in this set are called blank.
n-tuples
The number of blank in A×B is the number of elements in A multiplied by the number of elements in B. That is, |A×B| = |A|×|B|
elements
The product A×B×C= {(a,b,c)|a∈A b∈B, c∈C}. The elements in this set are called blank.
ordered triples
If s and t are two strings, then the blank of s and t (denoted st) is a longer string obtained by putting s and t together. If s = 010 and t = 11, then st = 01011. It is also possible to concatenate a string and a single symbol: t0 = 110. Concatenating any string x with the empty string gives back x: xλ = x.
concatenation
Two sets, A and B, are said to be blank if their intersection is empty (A ∩ B = ∅)
disjoint
A sequence of sets, A1, A2, …, An, is blank if every pair of distinct sets in the sequence is disjoint (i.e., Ai ∩ Aj = ∅ for any i and j in the range from 1 through n where i ≠ j).
pairwise disjoint
A blank of a non-empty set A is a collection of non-empty subsets of A such that each element of A is in exactly one of the subsets. A1, A2, …,An is a partition for a non-empty set A if all of the following conditions hold:
For all i, Ai ⊆ A.
For all i, Ai ≠ ∅
A1, A2, …,An are pairwise disjoint.
A = A1 ∪ A2 ∪ … ∪ An
partition
A blank of a non-empty set A is a collection of nonempty sets whose union is all of A and each pair in the collection is pairwise disjoint. Hint: Think of a partition as chopping the set up into disjoint pieces.
partition
A blank f that maps elements of a set X to elements of a set Y, is a subset of X × Y such that for every x ∈ X, there is exactly one y ∈ Y for which (x, y) ∈ f.
function
f: X → Y is the notation to express the fact that f is a function from X to Y. The set X is called the blank of f, and the set Y is the blank of f. The fact that f maps x to y (or (x, y) ∈ f) can also be denoted as f(x) = y.
domain
target
In an blank for a function f, the elements of the domain X are listed on the left and the elements of the target Y are listed on the right. There is an arrow from x ∈ X to y ∈ Y if and only if (x, y) ∈ f.
arrow diagram
For function f: X → Y, an element y is in the blank of f if and only if there is an x ∈ X such that (x, y) ∈ f. Expressed in set notation:
Range of f = { y: (x, y) ∈ f, for some x ∈ X }
The range of f is a subset of the target but the range is not necessarily equal to the target.
range
blank are functions whose pairing relationship is defined by an algebraic set of operations. For example, f : R → R where f(x) = x2 -2x + 3.
Algebraic functions
In an arrow diagram, domain elements are on the left, target elements on the right. An arrow for each domain element points to a target element.
In an arrow diagram for f, each element in the domain has exactly blank leaving it. If not, then it is not a function
one arrow
Two functions, f and g, are blank if f and g have the same domain and target, and f(x) = g(x) for every element x in the domain. The notation f = g is used to denote the fact that functions f and g are equal.
equal
The blank maps a real number to the nearest integer in the downward direction.
floor function
The blank rounds a real number to the nearest integer in the upward direction.
ceiling function
A function f: X → A is blank or one-to-one if x1 ≠ x2 implies that f(x1) ≠ f(x2). That is, f maps different elements in X to different elements in A.
injective
A function f: X → A is blank or onto if the range of f is equal to the target A. That is, for every a ∈ A, there is an x ∈ X such that f(x) = a.
surjective
A function is blank if it is both injective and surjective. A bijective function is called a bijection. A bijection is also called a one-to-one correspondence.
bijective
A function f: X → Y is blank if there exists a function g with domain Y and range X with the property
f(x) = y ⇔ g(y) = x.
invertible
If a function f: X → Y is a bijection, then the blank of f is obtained by exchanging the first and second entries in each pair in f. The inverse of f is denoted by f-1:
f-1 = { (y, x) : (x, y) ∈ f }.
inverse
Some functions do not have an inverse. A function f: X → Y has an inverse if and only if reversing each pair in f results in a blank from Y to X. f-1 is a well-defined function if every element in Y is mapped to exactly one element in X.
well-defined function
A function f has an inverse if and only if f is a blank.
bijection
The inverse of a bijection f can also be expressed in function notation. If f is a bijection from X to Y, then for every x ∈ X and y ∈ Y,
f(x) = y if and only if blank
Therefore, the value of f^-1(y) is the unique element x ∈ X such that f(x) = y. If f-1 is the inverse of function f, then for every element x ∈ X, f-1(f(x)) = x.
f^-1(y) = x.
If a function with an inverse has a finite domain, you can find the inverse by swapping the order in all the blank.
ordered pairs
Inverses for functions on infinite domains can be solved blank.
algebraically
To algebraically solve for an inverse, use the following algorithm:
.
1) Replace f(x)
with y
2)Interchange x
and y
3. Solve for y
4.Replace y
with f^-1(x)
The process of applying a function to the result of another function is called blank.
composition
f and g are two functions, where f: X → Y and g: Y → Z. The blank, denoted g ο f, is the function (g ο f): X → Z, such that for all x ∈ X, (g ο f)(x) = g(f(x)).
composition of g with f
The blank always maps a set onto itself and maps every element onto itself.
The blank on A, denoted IA: A → A, is defined as IA(a) = a, for all a ∈ A.
identity function
If a function f from A to B has an blank, then f composed with its inverse is the identity function. If f(a) = b, then f-1(b) = a, and (f-1 ο f)(a) = f-1(f(a)) = f-1(b) = a.
Let f: A → B be a bijection. Then f-1 ο f = IA and f ο f-1 = IB.
inverse
The exponential function expb:R → R+ is defined as: blank
where b is a positive real number and b ≠ 1.
expb(x) = b^x
The parameter b is called the blank in the expression bx.
base of the exponent
The input x to the function bx is called the blank.
exponent
blank exponents are combined by multiplying the exponents.
Nested
When you multiply powers with the same base, add the exponents
Product of powers rule
When the power is raised to a power, multiply exponents.
Power rule I
When you divide powers with the same base, subtract the exponents
Quotient rule
When you raise a product to a power, raise each factor with a power
Power of a product rule
The exponential function is one-to-one and onto, and therefore has an blank.
inverse
The logarithm function is the inverse of the exponential function. For real number b > 0 and b ≠ 1, logb:R+ → R is defined as: blank
The parameter b is called the base of the logarithm in the expression logb y.
b^x = y == logby = x
A function f is said to be blank if whenever x1 < x2, then f(x1) < f(x2
strictly increasing
A function f is said to be blank if whenever x1 < x2, then f(x1) > f(x2).
strictly decreasing
The inverse function to the exponential is the blank.
logarithmic function
The log of a product is equal to the sum of the log of the first base and the log of the second base
Log product rule
The log of a quotient is equal to the difference of the logs of the numerator and denominator
Log quotient rule
The log of a power is equal to the power times the log of the base
Log power rule
The log of a new base is the log of the new base divided by the log of the old base in the new base
Change of base formula
The exponential and the logarithmic function are both blank functions.
strictly increasing
