Vocab

Question 1

set

Answer 1

A collection of things called elements or members.

Question 2

irrational numbers

Answer 2

A number that cannot be written in the form m/n with m and n both integers.

Question 3

rational numbers

Answer 3

Numbers, which are ratios of integers with nonzero denominators. Rational numbers have decimal expansions which terminate or repeat.

Question 4

integers

Answer 4

The set of natural numbers, their negatives, and zero.

Question 5

natural numbers

Answer 5

All positive whole numbers excluding zero.

Question 6

complex numbers

Answer 6

Numbers, which have the form a+bi.

Question 7

Sets A and B are equal

Answer 7

A = B, if and only if A and B contain the same elements or neither set contains any element.

Question 8

subset

Answer 8

Set A is a subset of set B, if and only if every element of A is an element of B.

Question 9

power set

Answer 9

P(A), is the set of all subsets of A: P(A) = {B | B subset A}

Question 10

union of sets A and B

Answer 10

A U B, is the set of elements in A or in B (or in both).

Question 11

intersection of A and B

Answer 11

A ∩ B, is the set of elements that belong to both A and B.

Question 12

set difference of sets A and B

Answer 12

A \ B, is the set of elements of A that are not in B.

Question 13

symmetric difference of sets A and B

Answer 13

A ⨁ B, is the set of elements in A or in B, but not in both.

Question 14

Cartesian product of A and B

Answer 14

The set A x B = {(a,b) | a∈A,b∈B}

Question 15

binary relation from A to B

Answer 15

1) Is a set
2) A x B = {(a,b) | a∈A,b∈B}
3) is a subset of A x B

Question 16

binary relation on A (from A to A)

Answer 16

1) Is a set
2) A x A = {(a,b) | a∈A, b∈A}
3) is a subset of A x A

Question 17

reflexive relation

Answer 17

Given R on set A: If (a,a)∈R for ALL a∈A,then the relation is reflexive.

Question 18

symmetric relation

Answer 18

If (a,b) ∈ R, then (b,a) ∈ R.

Question 19

antisymmetric relation

Answer 19

If (a,b) ∈ R and (b,a) ∈ R, then a = b.

Question 20

transitive relation

Answer 20

A binary relation R on a set A is transitive if and only if a, b, c ∈ A, and both (a,b) and (b,c) ∈ R, then (a,c) ∈ R.

Question 21

equivalence relation on set A

Answer 21

A binary relation R on A that is reflexive, symmetric, and transitive. You must prove:

1) a ~ a for all a ∈ A,
2) if a ∈ A, and b ∈ A, and a ~ b, then b ~ a, and
3) if a, b, c ∈ A, and both a ~ b and b ~ c, then a ~ c.

Question 22

equivalence class

Answer 22

The group into which an equivalence relation divides the underlying set. The equivalence class of an element is the collection of all things related to it.
The equivalence class of element a ∈ A is the set a ̅ = {x ∈ A | x ~ a}.

Question 23

quotient set of A mod ~

Answer 23

Denoted A/~, it is the set of all equivalence classes.

Question 24

x ~ a or a~x?

Answer 24

An equivalence relation is symmetric, so it doesn’t matter. The set of things related to a is the same as the set of things to which a is related.

Question 25

Set A and set B are disjoint

Answer 25

A⋂B=∅

Question 26

equivalence relations

Answer 26

Equivalence classes of any equivalence relation divide A into disjoint (that is, non overlapping) subsets that cover the entire set, just like the pieces of a jigsaw puzzle. We say that the equivalence classes “partition” A or “form a partition of” A.

Question 27

partition of set A

Answer 27

A collection of disjoint nonempty subsets of A whose union is A.

Question 28

cells

Answer 28

Also blocks, are the disjoint sets of a partition. The cells are said to partition A.

Question 29

partial order

Answer 29

(≤) - A binary relation on real numbers that possesses the properties of:
reflexivity - a ≤ a for all a ∈ A.
antisymmetry - If a, b ∈ A, a≤b and b≤a, then a=b, and
transitivity - If a, b, c ∈ A, a≤b and b≤c, then a≤c.

Question 30

partial order set

Answer 30

poset - for short, is a pair (A, ≤) where ≺ is a partial order on the set A.

Question 31

total order

Answer 31

When ≤ is a partial order on a set A and every two elements of A are comparable.

Question 32

totally ordered set

Answer 32

The pair (A, ≤) in a total order.

Question 33

maximum poset element

Answer 33

The element a of poset (A, ≤), iff b≤a for every b ∈ A.

Question 34

minimum poset element

Answer 34

The element a of poset (A, ≤), iff a≤b for every b ∈ A.

Question 35

maximal poset element

Answer 35

The element a of poset (A, ≤), iff, if b ∈ A and a≤b, then b=a.

Question 36

minimal poset element

Answer 36

The element a of poset (A, ≤), iff, if b ∈ A and b≤a, then b=a.

Question 37

greatest lower bound(∧)

Answer 37

glb - (A, ≤) is a poset, the element g iff a,b,g ∈ (A, ≤), and
1 - g≤a, g≤b, and
2 - if c≤a, and c≤b for some c ∈ A, then c≤g.

Question 38

least upper bound (⋁)

Answer 38

lub - (A, ≤) is a poset, the element t iff a,b,t ∈ (A, ≤), and
1 - a≤t, b≤t, and
2 - if a≤c, and b≤c for some c ∈ A, then t≤c.

Question 39

lattice

Answer 39

A poset (A, ≤) in which every two elements have a greatest lower bound in A and a least upper bound in A.

Question 40

sequence

Answer 40

A function whose domain is some infinite set of integers (often N) and whose range is a set of real numbers.

Question 41

terms of the sequence

Answer 41

Then numbers in the list, the range of the function.

Question 42

recurrence relation

Answer 42

A relation which defines one member of the sequence in terms of a previous one.

Question 43

arithmetic sequence

Answer 43

With first term a and common difference d, is the sequence defined by:
a_1 = a,
and for k≥1,
a_(k+1) = ra_k

Question 44

geometric sequence

Answer 44

A sequence of numbers in which each term is determined by multiplying the previous term by a fixed number.

With the first term a and common ratio r, it is the sequence defined by
a_1 = a,
and for k≥1,
a_(k+1) = a_k + d

Question 45

common ratio

Answer 45

The fixed number that each term in a geometric sequence is multiplied by to get the next term in the sequence.

Question 46

generating function

Answer 46

A polynomial that “goes on forever,” that is, and expression of the form:
f(x)=a_0 + a_1 x + a_2 x^2 + a_n x^n + ⋯

Question 47

The union |A∪B| =

Answer 47

|A| + |B| - |A∩B|

Question 48

The intersection |A∩B|≤min⁡{|A|,|B|}

Answer 48

the minimum of |A| and |B|

Question 49

Only A |A \ B|

Answer 49

= |A| - |A∩B| ≥ |A| - |B|

Question 50

Not A |A^C |

Answer 50

= |U| - |A|

Question 51

A + B but not A&B |A⨁B|

Answer 51

=|A∪B| - |A∩B| = |A| + |B| - 2|A∩B| = |A\B|+|B\A|

Question 52

AxB |AxB|

Answer 52

= |A| x |B|

Question 53

mutually exclusive

Answer 53

The two events can not happen at the same time.

Question 54

n!

Answer 54

Denotes the product of all the natural numbers from 1 to n (inclusive).

n! = n(n-1)(n-2) … (3)(2)(1)

Question 55

P(n,r)

Answer 55

Denotes that product of the first r factors of n!

P(n,r) = (n(n-1)(n-2) … (n-r+1))/(n-r)!

Question 56

permutation of a set

Answer 56

of distinct symbols is an arrangement of them in a line in some order.

Question 57

binomial coefficient
n
r

Answer 57

= n!/(r!(n-r)!

Question 58

combination of a set

Answer 58

of objects is a subset of them. A subset of r objects is called an r-combination or a combination of objects taken r at a time.

Question 59

The distinction between permutation and combination

Answer 59

is the distinction between order and selection. A permutation takes order into account; a combination involves only selection.