MMW Deck 2

Language

A systematic way of communication with other people using sounds or conventions symbols.

Language (in discipline)

A system of words used in a particular discipline.

Mathematical language

The system used to communicate mathematical ideas.

Mathematical language

Consists of some natural language using technical terms (mathematical terms) and grammatical conventions that are uncommon to mathematical discourse, supplemented by a highly specialized symbolic notation for mathematical formulas.

Mathematical notation

Used for formulas has its own grammar and is shared by mathematicians anywhere in the globe.

Precise

Characteristic of mathematical language wherein it can make very fine distinctions among a set of symbols.

Concise

Characteristic of mathematical language wherein it can briefly express long sentences.

Powerful

Characteristic of mathematical language wherein it gives upon expressing complex thoughts.

Galileo Galilei

Said 'Mathematics is the language in which God has written the universe.'

Mathematical expression

A finite combination of symbols that is well-defined according to rules that depend on the context.

Symbols

Can designate numbers, variables, operations, functions, brackets, punctuations, and groupings to help determine the order of operations and other aspects of mathematical syntax.

Expression

A correct arrangement of mathematical symbols used to represent the object of interest.

Expression

It does not contain a complete thought, and it cannot be determined if it is true or false.

Mathematical sentence

Makes a statement about two expressions, either using numbers, variables, or a combination of both.

Mathematical sentence

Can also use symbols or words like equals, greater than, or less than.

Conventions

Mathematical languages have conventions, and it helps individuals distinguish between different types of mathematical expressions.

Mathematical convention

A fact, name, notation, or usage which is generally agreed upon by mathematicians. Example: principle of PEMDAS.

Conventions (mathematicians)

Mathematicians abide by conventions to be able to understand what they have written without constantly having to redefine basic terms.

Conventional

Almost all mathematical names and symbols are conventional.

Set theory

The branch of mathematics that studies sets or the mathematical science of the infinite.

Set

A well-defined collection of objects; the objects are called the elements or members of the set.

Element of a set

The symbol ∈ is used to denote that an object is an element of a set.

Not an element

The symbol ∉ denotes that an object is not an element of a set.

x/x

means "x is x such that"

Roster Method

set is comprised of digits. Also known as Listing Method.

Rule Method

set is comprised of words. Also known as Set-Builder Notation.

Finite set

is a set whose elements are limited or countable, and the last element can be identified. | Example: A = {1, 2, 3, 4}

Infinite set

is a set whose elements are unlimited or uncountable, and the last element cannot be specified. | Example: B = {1, 2, 3, 4, …}

Unit set

is a set with only one element. (or singleton set) | Example: C = {2}

Empty set

it is denoted by the symbol ∅ or {. } (or null set) | Example: D = { }

Joint Sets

Common elements (at least one) | Example: A = {x/x is a real number, 1<x<7}, B = {x/x is an even number less than 12}

Disjoint Sets

No common elements | Example: The set of even numbers, The set of odd numbers.

Universal set

is the totality of all elements in the area of interest. (denoted by U) | Example: The set of all counting numbers. In symbol, U = {1, 2, 3, …}

Cardinal number

of a set is the number of elements or members in the set, the cardinality of set A is denoted by n(A). (Sirug, 2018) | Example: A = {1, 2, 3}, Set A has 3 elements. Hence, n(A) = 3

Equivalent Sets

With the same cardinal number. | Example: A = {1, 2, 3} Here n(A) = 3, B = {p, q, r} Here n(B) = 3. Therefore, A ↔ B

Equal sets

With same elements | Example: A = {j, a, y}, B = {y, a, j}

Subsets

If A and B are sets, A is called subset of B, written A⊆B, if and only if, every element of A is also an element of B.

Proper subset

A is a proper subset of B, written A⊂B, if and only if, every element of A is in B but there is at least one element of B that is not in A.

Power set of S

denoted by P(S) is the set of all the subsets of a set.

Union of Sets

The union of A and B, denoted by A∪B, is the set of all elements x in U such that x is in A or x is in B.

Intersection of Sets

Let A and B be subsets of a universal set U. The intersection of A and B, denoted by A∩B, is the set of all elements x in U such that x is in A and x is in B.

Complement of a Set

The complement of A (absolute complement of A), denoted by A', is the set of all elements x in U such that x is not in A.

Difference of Two Sets

The difference of A and B (or relative complement of B with respect to A), denoted by A ~ B, is the set of all elements x in U such that x is in A and x is not in B.

Cartesian Product

The product of sets A and B, written A x B.