Introduction to University Mathematics

Question 1

Natural Number

Answer 1

A member of the sequence 0,1,2,… formed by starting with 0 and succesively adding 1.

Question 2

Ordering of natural numbers

Answer 2

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).

Question 3

The principle of mathematical induction

Answer 3

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.

Question 4

Addition of the natural numbers

Answer 4

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

Question 5

The binomial coefficient

Answer 5

nCk = (n!)/[(n-k)!(k!)]

Question 6

Set

Answer 6

A collection of objects.

Question 7

Elements

Answer 7

The objects of a set.

Question 8

Subset

Answer 8

A is a subset of S if every element in A is also in S.

Question 9

Proper subset

Answer 9

A subset, but the two sets are not equal.

Question 10

Empty set

Answer 10

The set with no elements.

Question 11

Power set

Answer 11

The set of all subsets.

Question 12

The union of subsets A and B of S

Answer 12

The set of elements of S such that they are also a member of A or B.

Question 13

The intersection of subsets A and B of S

Answer 13

The set of elements of S such that they are also a member of A and B.

Question 14

The complement of a subset A of S

Answer 14

The set of elements of S such that they are not a member of A.

Question 15

The set difference of A and B

Answer 15

The set of elements of S such that they are a member of A but not B.

Question 16

Disjoint

Answer 16

A and B are disjoint if the intersection of A and B is the empty set.

Question 17

Relation

Answer 17

A relation on sets A and B is a subset of AxB. If (a,b) are members of R, we write aRb.

Question 18

Reflexive relation

Answer 18

Given a set S and a relation R on S, R is reflexive if xRx for all x in S.

Question 19

Symmetric relation

Answer 19

Given a set S and a relation R on S, R is symmetric if xRy implies yRx for all x and y in S.

Question 20

Anti-symmetric relation

Answer 20

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.

Question 21

Transitive

Answer 21

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.

Question 22

Partial order relation

Answer 22

A relation which is reflexive, anti-symmetric and transitive.

Question 23

Total order relation

Answer 23

A relation which is reflexive, anti-symmetric and transitive and where for all x and y in S, either xRy or yRx.

Question 24

Equivalence relation

Answer 24

A relation which is reflexive, symmetric and transitive.

Question 25

An equivalence class of x

Answer 25

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.

Question 26

Partition of a set

Answer 26

The partitions of a set S is a collection of non-empty disjoint subsets whose union is S.

Question 27

A function

Answer 27

For sets X and Y, a function, f: X -> Y assigns a value of Y, f(x), for all x in X.

Question 28

Domain

Answer 28

The set which a function acts on.

Question 29

Codomain

Answer 29

The set which the function acts onto.

Question 30

Well defined function

Answer 30

For all elements in the domain, there is a unique value of f(x) in the codomain.

Question 31

Image or range

Answer 31

The subset of the codomain generated by the function when acting on each value in the domain.

Question 32

Preimage

Answer 32

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.

Question 33

Restriction of a function

Answer 33

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.

Question 34

Identity map

Answer 34

A function X -> X such that id(x) = x for all x in X.

Question 35

Define an injective function

Answer 35

f: X -> Y is injective if f(x1) = f(x2) implies that x1 = x2.

Question 36

Define a surjective function

Answer 36

f: X -> Y is surjective if for all y in Y, there exists x in X such that f(x) = y.

Question 37

Define a bijective function

Answer 37

A function which is both injective and surjective.

Question 38

Define an invertible function

Answer 38

A function f: X -> Y is invertible if there exists g: Y -> X such that fg = id(Y) and gf = id(X).