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Math 290 Review
Definitions for Final Exam
58
Mathematics
Undergraduate 2
04/19/2011
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Term
A set A is called a subset of a set B if...
 Definition
...every element of A also belongs to B.
Term
A set A is a proper subset of a set B if...
 Definition
...A⊆B but A≠B.
Term
The power set of A is...
 Definition
...the set consisting of all subsets of a given set A.
Term
The union of two sets A and B is...
 Definition

...the set of all elements belong to A or B.

A∪B={x : x ∈ A or x ∈ B}.
Term
The intersection of two sets A and B is...
 Definition

...the set of all elements belonging to both A and B.

A∩B={x : x ∈ A and x ∈ B}
Term
A and B are said to be disjoint if...
 Definition

...A and B have no elements in common.

A∩B=∅
Term
The difference of two sets A and B is defined as...
 Definition
...A−B={x : x ∈ A and x ∉ B}.
Term
For a set A, its complement is...
 Definition
...Ā=U−A={x : x ∈ U and x ∉ A} where U is the universal set.
Term
A parition of A is...
 Definition
...a collection S of nonempty subsets of A such that every element of A belongs to exactly one subset of S.
Term
The Cartesian Product of two sets A and B is...
 Definition

...the set consisting of all ordered pairs whose first coordinate belongs to A and whose second coordinate belongs to B.

A×B={(a,b): a ∈ A and b ∈ B}
Term
A statement is...
 Definition
...a declarative sentence or asseration that is true or false (but not both).
Term
An open sentence is...
 Definition
...a statement that involves a variable. Once the variables are decided, then the statement is true or false.
Term
For statements P and Q the disjunction of P and Q is...
 Definition
...the statement P or Q and is true if either P or Q is true, false otherwise, and is denoted denoted P ∨ Q.
Term
For statements P and Q the conjunction of P and Q, is...
 Definition
... the statment P and Q and is true if P and Q are both true, false otherwise, and is denoted P ∧ Q.
Term
For statements P and Q, the implication...
 Definition
...is the statement If P, then Q and false when P is true and Q is false, true otherwise. It is denoted by P ⇒ Q.
Term
For statements P and Q, the biconditional of P and Q is...
 Definition
...the statement P if and only if Q and is true if P and Q are both true or both false, false otherwise. It is denoted P ⇔ Q.
Term
A compound statement S is called a tautology if...
 Definition
...it is true for all possible combinations of truth values of the component statements that comprise S.
Term
A compound statement S is a contradiction if...
 Definition
...it is false for all possible combinations of truth values of the component statements that are used to form S.
Term
Compound statements R and S are logically equivalent if...
 Definition
...R and S have the same truth values for all combinations of truth values of their component statements. It is denoted R ≡ S.
Term
The phrase "for every" is referred to as...
 Definition
...the universal quantifier and is denoted by the symbol ∀.
Term
Each of the phrases "there exists", "there is", "for some", and "for at least one" is referred to as...
 Definition
...an existential quantifier and is denoted by the symbol ∃.
Term
A statement is trivially true if...
 Definition
...the conclusion is always true regardless of the hypothesis.
Term
A statement is vacuously true if...
 Definition
...the hypothesis is always false.
Term
For statements P and Q, the contrapositive of the implication P ⇒ Q is...
 Definition
...the implication (∼Q) ⇒ (∼P).
Term
For integers a and b with a≠0, we say that a divides b if...
 Definition

...there is an integer c such that b=ac.

Denoted a|b
Term
For integers a and b, b is a multiple of a if...
 Definition
...a divides b.
Term
For integers a, b, and n≥2, we say that a is congruent to b modulo n, written a ≡ b (mod n) if...
 Definition
...n|(a−b).
Term
Triangle Inequality
 Definition
|x+y|≤|x|+|y|
Term
A number m∈A is called a least element of A if...
 Definition
...x≥m for every element x∈A.
Term
A nonempty set S is said to be well-ordered if..
 Definition
...every nonempty subset of S has a least element.
Term
A relation R from A to B is...
 Definition
...a subset of A×B.
Term
The domain of R is..
 Definition

...the subset of A defined by

dom R = {a∈ A ; (a,b)∈ R for some b∈ B}.
Term
The range of R, denoted by ran R, is..
 Definition

...the subset of B defined by

ran R = {b∈ B ; (a,b)∈ R for some a∈ A}.
Term
A relation R on set A is reflexive if...
 Definition
...there exists (a,a)∈ R for all a∈ A.
Term
A relation R on set A is symmetric if...
 Definition
...(a,b)∈R then (b,a)∈R for all a,b∈A.
Term
A relation R on set A is transitive if...
 Definition
...(a,b),(b,c)∈R implies (a,c)∈R for all a,b,c∈A.
Term
A relation R on set A is irreflexive if...
 Definition
...(a,a)∉R for all a,a∈A.
Term
A relation R on set A is anti-symmetric if...
 Definition
...(a,b),(b,a)∈R implies a=b for all a,b∈A.
Term
A relation R on A is called an equivalence relation if...
 Definition
...R is reflexive, symmetrical, and transitive.
Term
For an equivalence relation R defined on a set A, and for a∈ A, the equivalence class of a is...
 Definition
...the set [a]={x∈ A : (x,a)∈ R}.
Term
A function, ƒ:A→B is...
 Definition
...a relation from A to B such that each element of A is related to exactly one element of B.
Term
A function ƒ:A→B is injective if...
 Definition
...(a1,b),(a2,b)∈ƒ implies a1=a2.
Term
A function ƒ:A→B is surjective if...
 Definition
...for all b∈B there exists a∈A such that (a,b)∈ƒ.
Term
A function ƒ:A→B is bijective if...
 Definition
...it is both injective and surjective.
Term
Let f:A→B and g:B→C both be functions.  Then the composition of f and g is...
 Definition
...(g ◦ f):A→C.
Term
For relation R from set A to set B, the inverse function from B to A is...
 Definition
...R-1={(b,a) : (a,b)∈R}.
Term
A set A is denumerable if...
 Definition
...there exists a bijection f:N→A.
Term
A set A is countable if...
 Definition
...it is finite or denumerable.
Term
A set A is said to have a smaller cardinality than a set B if...
 Definition
...there exits an injective function from A to B but no bijective function from A to B.
Term
Division Algorithm
 Definition
For positive numbers a and b, there exists unique integers q and r such that b=aq+r and 0≤r<a.
Term
For integers a and b, an integer of the form ax+by, where x,y∊Z, is called...
 Definition
...a linear combination of a and b.
Term
Two integers a and b, both not 0, are relatively prime if...
 Definition
...gcd(a,b)=1.
Term
Canonical Factorization
 Definition
An integer n≥2 can be uniquely expressed as a product of primes.
Term
A sequence (of real numbers) is...
 Definition
...a real valued function defined on the set of natural number.
Term
A sequence {an} of real numbers is said to converge to the real number L if...
 Definition
...for every real number ε>0, there exists a positive integer N such that if n is an integer with n>N, then |an−L|<ε.
Term
If a sequence {an} converges to L, then...
 Definition
...L is referred to as the limit of {an} and we write limn→∞an=L.
Term
For a function f:X→R with a∈X, the deleted neighborhood is...
 Definition
...the set of type (a-δ,a)∪(a+δ)=(a-δ,a+δ)-{a}⊆X for some positive real number δ.
Term
L is the limit of f(x) as x approaches a, written limx→af(x), if...
 Definition
...for every real number ε>0, there exists a real number δ>0 such that for every real number x with 0<|x−a|<δ, it follows that |ƒ(x)−L|<ε.
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