Introduction to University Mathematics Flashcards
Natural Number
A member of the sequence 0,1,2,… formed by starting with 0 and succesively adding 1.
Ordering of natural numbers
For two natural numbers m,n, we can write m ≤ n, meaning that there exists a natural number k such that m + k = n (or vice verse).
The principle of mathematical induction
For a family of statements indexed by the natural numbers, P(n), if we show that P(0) is true and that P(n) implies P(n+1), than P(n) is true for all n.
Addition of the natural numbers
For any m and n in the natural numbers, the following two axioms hold:
i) m + 0 = m
ii) m + (n + 1) = (m + n) + 1
The binomial coefficient
nCk = (n!)/[(n-k)!(k!)]
Set
A collection of objects.
Elements
The objects of a set.
Subset
A is a subset of S if every element in A is also in S.
Proper subset
A subset, but the two sets are not equal.
Empty set
The set with no elements.
Power set
The set of all subsets.
The union of subsets A and B of S
The set of elements of S such that they are also a member of A or B.
The intersection of subsets A and B of S
The set of elements of S such that they are also a member of A and B.
The complement of a subset A of S
The set of elements of S such that they are not a member of A.
The set difference of A and B
The set of elements of S such that they are a member of A but not B.
Disjoint
A and B are disjoint if the intersection of A and B is the empty set.
Relation
A relation on sets A and B is a subset of AxB. If (a,b) are members of R, we write aRb.
Reflexive relation
Given a set S and a relation R on S, R is reflexive if xRx for all x in S.
Symmetric relation
Given a set S and a relation R on S, R is symmetric if xRy implies yRx for all x and y in S.
Anti-symmetric relation
Given a set S and a relation R on S, R is anti-symmetric if xRy and yRx implies x = y for all x and y in S.
Transitive
Given a set S and a relation R on S, R is transitive if xRy and yRz implies that xRz for all x, y and z in S.
Partial order relation
A relation which is reflexive, anti-symmetric and transitive.
Total order relation
A relation which is reflexive, anti-symmetric and transitive and where for all x and y in S, either xRy or yRx.
Equivalence relation
A relation which is reflexive, symmetric and transitive.
An equivalence class of x
Given an equivalence relation ~ on a set S and an element x in S, the equivalence class is the set of y in S such that y ~ x.
Partition of a set
The partitions of a set S is a collection of non-empty disjoint subsets whose union is S.
A function
For sets X and Y, a function, f: X -> Y assigns a value of Y, f(x), for all x in X.
Domain
The set which a function acts on.
Codomain
The set which the function acts onto.
Well defined function
For all elements in the domain, there is a unique value of f(x) in the codomain.
Image or range
The subset of the codomain generated by the function when acting on each value in the domain.
Preimage
A preimage of B (a subset of the codomain) is a subset of the domain generated by the elements such that f(x) is in B.
Restriction of a function
Given a function f: X -> Y and a subset A of the domain, the restriction of f to A is a function A -> Y generated by f(x) for all x in A.
Identity map
A function X -> X such that id(x) = x for all x in X.
Define an injective function
f: X -> Y is injective if f(x1) = f(x2) implies that x1 = x2.
Define a surjective function
f: X -> Y is surjective if for all y in Y, there exists x in X such that f(x) = y.
Define a bijective function
A function which is both injective and surjective.
Define an invertible function
A function f: X -> Y is invertible if there exists g: Y -> X such that fg = id(Y) and gf = id(X).
