Mathematics and Language

What quality of language allows it to make fine distinctions?

Precise

What quality of language allows long sentences to be expressed in shorter ones?

Concise

What quality of language allows complex thoughts to be expressed with relative ease?

Powerful

It is a term that represents a quantity or operation.

Mathematical expression

It is a term that states a complete mathematical statement using a relation symbol.

Mathematical sentence

Is “a ÷ b” a mathematical expression or sentence?

Mathematical expression

Is “a ÷ b ≠ 9” a mathematical expression or sentence?

Mathematical sentence

What is the convention in mathematics for the order of operations?

PEMDAS

It is a collection of well-defined objects (elements)

Set

These are counting numbers: 1, 2, 3, 4, ...

Natural numbers (N)

These are natural numbers including zero: 0, 1, 2, 3, ...

Whole numbers (W)

These are whole numbers and their opposites: ..., −3, −2, −1, 0, 1, 2, 3,

Integers (Z)

These are numbers that can be expressed as a fraction of two integers, where the denominator is not zero

Rational numbers (P)

What are the 2 ways to describe a set?

Roster method and Set builder notation

Which method describes a set by directly listing its elements?

Roster method

Which method describes a set by stating a rule or common property of its elements?

Set-builder notation

It is a set with no elements.

Empty (null) set

What are the symbols for an empty (null) set?

∅ or { }

It is a set with one element.

Singleton

What is an example of a singleton set?

{5}

It is a set with a limited number of elements

Finite set

What is an example of a finite set?

{a, b, c}

It is a set with an unlimited number of elements

Infinite set

What is an example of an infinite set?

(1, 2, 3, ...}

These are sets that have the same elements.

Equal set

What is an example of equal sets?

{1, 2, 3} = {3, 2, 1}

Does the order of elements matter in equal sets?

No

These are sets that have the same number of elements

Equivalent sets

What is an example of equivalent sets?

{a, b, c} and {x, y, z}

What is the term for a set B in which every element of B is also an element of set A?

Subset

What symbol denotes that B is a subset of A?

B ⊆ A

What is the term for the list of all subsets of a set S?

Power set

What is the notation for the power set of S?

P(S)

What formula is used to determine the number of subsets of a set?

2ⁿ

In the formula 2ⁿ, what does n represent?

Number of elements in the set

What is a set B called if every element of B is in A, but at least one element of A is not in B?

Proper subset

What symbol denotes that B is a proper subset of A?

B ⊂ A

What is the set that contains all elements under discussion called?

Universal set

What is the number of elements in a set called?

Cardinality

What are the four basic set operations?

Difference, Intersection, Union, Complement

What is the notation for the cardinality of set A?

n(A)

What operation combines all elements that are in A or B?

Union (A ∪ B)

What operation identifies the elements common to both A and B?

Intersection (A ∩ B)

What operation identifies the elements in A but not in B?

Difference (A − B)

What operation identifies the elements that are not in A?

Complement (A′)

What is an ordered pair (x, y) in which x is the domain and y is the range called?

Relation

In the ordered pair (x, y), what does x represent?

Domain

In the ordered pair (x, y), what does y represent?

Range

What are the three ways a relation can be represented?

Table, mapping, graph

What is a relation in which each element of x is paired with exactly one element of y called?

Function

What method is used to determine whether a graph represents a function?

Vertical line test

What is drawn at any part of the graph when using the vertical line test?

A vertical line

What happens if a vertical line intersects only one point of the graph?

The relation is a function

What is the combining of functions called?

Composition of functions

What is a function mapping S × S to S called?

Binary operation

What do binary operations perform with operands?

Mathematical operations

What are examples of mathematical operations performed by binary operations?

Addition and subtraction

What is the notation for the function mapping S × S to S?

*

What are the five properties of binary operations?

Closure, Commutativity, Associativity, Identity elements, Inverse elements

Which property ensures that the result of a binary operation remains within the set?

Closure

Which property means that changing the order of operands does not change the result?

Commutativity

Which property means that changing the grouping of operands does not change the result?

Associativity

What is the closure property for addition and multiplication?

x + y ∈ R, Z and xy ∈ R, Z

What is the commutative property for addition and multiplication?

x + y = y + x and xy = yx

What is the associative property for addition?

x + (y + z) = (x + y) + z

What is the associative property for multiplication?

(xy)z = x(yz)

What are the identity elements for addition and multiplication?

0 for addition; 1 for multiplication

What is the identity property for addition?

x + 0 = x

What is the identity property for multiplication?

x · 1 = x

What is the inverse property for addition?

x + (−x) = 0

What is the inverse property for multiplication?

x · 1/x = 1

What is the additive inverse of x?

−x

What is the multiplicative inverse of x?

1/x