statements of definition

Question 1

principle of mathematical induction

Answer 1

suppose that for each positive integer n we have a statement P(n). If we prove the following two things:
(a) P(1) is true;
(b) for all n, if P(n) is true then P(n+1) is also true;
then P(n) is true for all positive integers n

Question 2

principle of mathematical induction ii

Answer 2

let k be an integer. suppose that for each integer n ≥ k we have P(n). if we prove the following two things
(a) P(k) is true;
(b) for all n ≥ k, if P(n) is true then P(n+1) is also true;
then P(n) is true for all integer n ≥ k

Question 3

strong principle of mathematical induction

Answer 3

If P is a set of integers such that:
(i) a is in P
(ii) if all integers k, with a ≤ k ≤ n are in P then the integer n + 1 is also in P,
then P = {x ∈ Z | x ≥ a} that is, P is the set of all integers greater than or equal to a.

Question 4

prime numbers

Answer 4

For each p ≤ sqrt(n), if p divides n, then n is prime.

Question 5

Plane graph and connectedness of a plane graph

Answer 5

When a connected graph can be drawn without an edges crossing, it is called planar. When a planar graph is drawn in this way, it divides the plane into regions called faces.

Question 6

division of integers

Answer 6

m/n (where m, n are integers) are called rational numbers (set Q)

Question 7

highest common factor

Answer 7

Let a, b belong to Z
a common factor of a and b is an integer that divides both a and b.
written as hcf(a,b)

Question 8

coprimality

Answer 8

Two numbers that have no common factor other than 1 and -1, though each number individually may not be prime, together they are coprime
Ex: 4 and 9
If a, b are coprime to each other, then there are integers s, t such that 1 = sa +tb

Question 9

congruence of integers

Answer 9

Let m be a positive integer.
For a, b belonging to Z, if m | b - a,
a ≡ b mod m
say a is congruent to b modulo m.

Question 10

multiplication principle (theorem 16.1)

Answer 10

Let P be a process which consists of n stages, and suppose that for each r, the rth stage can be carried out in ar ways. Then P can be carried out in a1a2…a’n ways.

Question 11

binomial and multinomial coefficients

Answer 11

(n r) called “n choose r”
(n!)/((r!)(n-r!))

Question 12

unions

Answer 12

union of A and B, written as A∪B is the set consisting of all elements that lie in either A or B (or both)

Question 13

intersections

Answer 13

intersection of A and B, written A∩B, is the set consisting of all elements that lie in both A and B

Question 14

cartesian products

Answer 14

cross product, where
AB= [(a1b1) (a1b2)]
(a2b1) (a2*b2)

Question 15

differences

Answer 15

a’n*b’n - ab = (a’n -a)b’n+a(b’n -b)

Question 16

Triangle Inequality

Answer 16

|a’n*b’n - ab| ≤
|a’n -a||b’n|+|a||b’n -b|

Question 17

partitions

Answer 17

Let n be a positive integer,
and let S={1, 2, …., n}
A partition of S is a collection of subsets S1, …, S’k such that each element of S lies in exactly one of these subsets. The partition is ordered if we take account of the order in which the subsets are written.

Question 18

equivalence relations

Answer 18

a relation is equivalent if
a, b, c belongs to S:
(i) a~a (this says ~ is reflexive)
(ii) if a~b then b~a (this says ~ is symmetric
(iii) if a~b and b~c then a~c (this says ~ is transitive)

Question 19

classes

Answer 19

Let S be a set and ~ an equivalence relation on S. For a ∈ S, define
cl(a)={s|s ∈ S, s~a}
cl(a) is the set of things that are related to a. The subset cl(a) is called an equivalence class of ~. The equivalence classes of ~ are the subsets cl(a) as a ranges over the elements of S.

Question 20

injective

Answer 20

Let f: S→T be a function
f is one-to-one if whenever s1, s2 ∈S with s1≠s2, then f(s1)≠f(s2); in other words, f is 1-1 if f sends different elements of S to different elements of T.

Question 21

surjective

Answer 21

Let f: S→T be a function
f is onto if the image f(S)=T; i.e. for every t∈T there exists s∈S such that f(s)=t

Question 22

bijective

Answer 22

a function is both onto and 1-1
(both surjective and injective)

Question 23

permutations

Answer 23

a bijective function from a set to itself
the number of permutations in S’n is n!

Question 24

countable sets

Answer 24

can be put into a one-to-one correspondence with the set of natural numbers
(injective to natural numbers)

Question 25

cardinality

Answer 25

the number of a sets elements, aka its size

Question 26

the powerset P(S)

Question 27

binary operations

Answer 27

A binary operation * on S is a rule which assigns to any ordered pair(a,b)(a,b ∈ S) an element a * b ∈ S. It is a function from S x S to S

Question 28

groups

Answer 28

A group (G, *) is a set G with a binary operation * satisfying the axioms

Question 29

subgroups

Answer 29

Let (G, *) be a group and let H be a subset of G. We say that H is a subgroup of (G, *) if H is itself a group under *

Question 30

order of an element and a group

Answer 30

Let G be a group, and let a ∈ G. The order of a, written o(a), is the smallest positive integer k such that a^k = e. If no such k exists, we say that a has infinite order and write o(a) = ∞

Question 31

cosets

Answer 31

For x ∈ G, Hx = {hx: h ∈ G} = {h(1)x,...hmx}

Question 32

derivative (first principles)

Question 33

critical number

Question 34

concavity

Question 35

inflection point

Question 36

dot product

Answer 36

a = {a1, a2, a3} b = {b1, b2, b3} a x b = a1b1 + a2b2 + a3b3

Question 37

cross product

Answer 37

multiply two vectors in three-dimensional space to get another vector that is perpendicular to both of the original vectors

Question 38

left/ lower riemann sums

Question 39

right/ upper riemann sums

Question 40

antiderivatives

Question 41

area between two curves

Question 42

length of a curve

Question 43

tangent and normal to a space curve

Question 44

order of a differential equation

Question 45

linearity of a differential equation

Question 46

homogeneity of a differential equation

Question 47

row echelon form

Question 48

reduced row echelon form

Question 49

consistency of a linear system

Question 50

transpose of a matrix

Question 51

symmetric matrices