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Math 13 Definitions
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Key Concepts:
Terms in this set (32)
proposition
a statement that is either true or false
tautology
logical expression that is always true, regardless of what the component statements might be.
contradiction
logical expression that is always false
Let n and p be integers. We say that n is divisible by p
if n = pk for some integer k
An integer ≥ 2 is prime
if the only positive integers it is divisible by are itself and 1.
A propositional function
is a family of propositions which depend on one or more
variables. The collection of allowed variables is the domain.
Let m and n be integers. We say that n divides m and write n|m if m is divisible by n if:
there exists some integer k such that m = kn
Let a and b be integers, and n a positive integer. We say that a is congruent to b modulo n and write
a≡b (modn)
if a and b have the same remainder upon dividing by n.
Let m and n be integers, not both zero. Their greatest common divisor gcd(m, n), is the
largest (positive) divisor of both m and n
A set A is finite
f it contains a finite number of elements: this number is the set's cardinality, written |A|
If A and B are sets such that every element of A is also an element of B, then we say that A is a
subset of B
Let A ⊆ U be a set. The complement of A is the set
A^c = { x ∈ U : x ∈/ A } .
If B ⊆ U is some other set, then the complement of A relative B is
B \ A = { x ∈ B : x ∈/ A } .
Let A and B be sets. A function from A to B is
a rule f that assigns one (and only one) element of B to each element of A.
The domain of f, written dom(f), is the set A. The codomain of f is the set B.
The range or image of f , written range( f ) or Im( f ), is the subset of B consisting of all the elements assigned by the rule f .
A function f : A → B is 1-1 (one-to-one), injective, or an injection if
if it never takes the same value twice. Equivalently:
∀ a 1 , a 2 ∈ A , f ( a 1 ) = f ( a 2 ) =⇒ a 1 = a 2 .
f : A → B is onto, surjective, or a surjection if
it takes every value in the codomain: i.e.,
B = range( f )
∀b ∈ B, ∃a ∈ A such that f (a) = b.
A set of real numbers A is well-ordered if
every non-empty subset of A has a minimum element.
If q ∈ N≥2 is not prime, then it is said to be composite:
∃a, b ∈ N≥2 such that q = ab.
Let A and B be sets. The Cartesian product of A and B is the set
A × B = {(a, b) : a ∈ A and b ∈ B}.
The power set of A is the set P(A) of all subsets of A. That is,
P(A) = {B : B ⊆ A}.
Otherwise said: B ∈ P(A) ⇐⇒ B ⊆ A.
Let A and B be sets. A (binary) relation R from A to B is a set of ordered pairs
a set of ordered pairs:
R ⊆ A × B.
If R⊆A×B is a relation, then its inverse R−1 ⊆ B×A is
the set R−1 ={(y,x) ∈ B×A: (x,y)∈R}.
Let R ⊆ A × B be a relation from A to B. The domain and range of R are the sets
dom(R) = {a ∈ A : (a, b) ∈ R for some b ∈ B}, range(R) = {b ∈ B : (a, b) ∈ R for some a ∈ A}.
A function from A to B is a relation f ⊆ A × B satisfying the following conditions:
1. dom(f)=A
2. (a,b1)(a,b2)∈f =⇒b1=b2.
Reflexivity
∀x ∈ A, x R x
Symmetry
∀x,y∈A, xRy =⇒ yRx
Transitivity
∀x,y,z∈A, xRy and yRz =⇒ xRz
An equivalence relation is a relation ∼ which is
reflexive, symmetric and transitive
Let ∼ be an equivalence relation on a set X. The equivalence class of
an element x ∈ X is the set
[x] = {y ∈ X : y ∼ x}.
Let X be a set and {An : n ∈ I} be a collection of non-empty subsets An ⊆ X. We say that X is partitioned
1. X = ∪An.
2. If Am ≠ An, then Am ∩ An = ∅.
The cardinalities of two sets A, B are denoted |A| and |B|. We compare cardinalities as follows:
• |A| ≤ |B| ⇐⇒
• |A| = |B| ⇐⇒
∃f : A → B injective.
∃f : A → B bijective.
A set A is uncountable if
|A| > א0, that is if there exists an injection f : N → A but no bijection from
g : N → A.
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