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Chapter 2. Mathematical Language AND Symbols
Word Processing With Document Production (Cavite State University)
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Chapter 2
Mathematical Language and Symbols
Language – the system of words, signs and symbols which people use to express ideas, thoughts and
feelings.
Mathematical Language – the system used to communicate mathematical ideas.
A. Characteristic of Mathematical Language
According to Jamison (2000) the use of language in mathematics differs from the language of
ordinary speech in three important ways.
 First, mathematical language is non-temporal.
 Second, mathematical language is devoid of emotional content
 Third, mathematical language is precise.
B. Mathematical Expression and Sentences
Math can be words, expressions and sentences that can help students explain what they
think.
But before we tackle the Mathematical expression and Sentences , let try to translate the
mathematics word to expression.
1.
2.
3.
4.
5.
6.
The sum of twice a number and six. 2x + 6
The sum of a number and 9. X + 9
The difference of a number and twenty-one. X – 21
The product of six and a number. 6a or 6 x a
The product of two and the sum of three and four. 2(3+4)
Four is added to the product of eight and a number. 8a + 4
Mathematical Expression
- Consist of term
- Separated with other term with either plus or minus.
- A single term may contain an expression in parenthesis or other grouping symbols.
- Mathematical expression may consist numerical coefficient, literal coefficient and
constant.
Mathematical Sentence
-
Combines two mathematical expression using a comparison operator
These expressions either use numbers, variables or both.
The comparison operator include equal, not equal, greater than, greater than or equal
to, less than and less than or equal to.
Examples of Equation.
Example of Inequality
4x + 3 = 19
15x -5 < 3y
6y – 5 = 55
18 > 16.5
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Open Sentence - means that it uses variables, meaning that it is not known whether or not the
mathematical sentence is true or false.
Closed Sentence – that known to be either true or false. It can be a TRUE CLOSED SENTENCE and
FALSE CLOSED SENTENCE.
Example of open sentence:
2xy < 3y
18 w > 16.5
3 (m + n) = 100
8ab –c = 1
x +y = 5
Example of TRUE CLOSED SENTENCE:
2 ( x + y ) = 2x + 2y
25 = 5
8c – c = 7c
Example of FALSE CLOSED SENTENCE;
9 is an even number
10 – 1 = 8
The square root of 4 is 1.
C. Conventions in the Mathematical Language
There are many symbols in mathematics and most are used as a precise form of shorthand. We
need to be confident when using these symbols, and to gain that confidence we need to understand their
meaning. To understand their meaning there are two things to help us –


D.
1.
2.
3.
4.
CONTEXT - this is the context in which we are working, or the particular topics being
studied,
CONVENTION - where mathematicians and scientists have decided that particular
symbols will have particular meaning.
FOUR BASIC CONCEPTS
SETS
FUNCTIONS
RELATIONS
BINARY OPERATIONS
1. SETS – a well-defined collection of distinct object and is denoted by an uppercase letter
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An object that belongs to a set is called an ELEMENT or MEMBER and it is usually denoted by
lower case letter. The symbol “” denotes a membership while “” denotes non- membership to a set.
Example
X  A read as x is an element of A.
1.1 Ways of Describing a Set


ROSTER/ TABULAR METHOD
- Method in which the elements in the given set are listed or enumerated, separated
by a comma, inside a pair or braces.
RULE/ DESCRIPTIVE METHOD
- Method in which the common characteristics of the elements are defined. This
method uses set builder notation where x is used to represent any element of the
given set.
Example.
a. The distinct letters in the words “mathematics”
Let A be the set of distinct etters of the word “mathematics”
Roster form : A = {m, a, t, h, e, m, a, t, I, c, s}
Rule form : A = {x∣x is the distinct letters in the words “mathematics” }
b. The colors of the rainbow
Let B be the set of the colors of the rainbow
Roster form : A = {red, orange, yellow, green, blue , indgo, violet }
Rule form : A = {x∣x is the colors of the rainbow }
c.
An even prime number
Let C be the set of the even prime number
Roster form : A = { 2 }
Rule form : A = {x∣x is the even prime number}
1.2 KIND OF SET
 EMPTY/ NULL/ VOID SET – set that has no elements, denoted by Ø or by a pair of braces with
no element inside.
Ex. The set of numbers in the English Alphabet
 FINITE SET - a set with a countable number of elements.
Ex. The set of letters in the English Alphabet
 INFINITE SET – a set has uncountable number of element
Ex. The set of counting numbers
 UNIVERSAL SET - the totality of all the elements of the sets under consideration, denoted by U.
Ex. The set of real numbers.
1.3 RELATIONSHIPS OF SETS
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
EQUAL SETS – set with same elements

EQUIVALENT SETS – set with the same number of elements.
Example: Let A = { x∣x is the distinct letter of the word “ read”}
And Let B = { x∣x is the distinct letter of the word “ dear”}
Therefore , sets A and B are equal sets, denoted by A = B since both sets have the elements a, d,
e, and r.
Since they have both four elements they are also consider as equivalent sets.
Remember: It can be noted that equal sets are equivalent sets, however not all equivalent sets are equal
sets.

JOINT SETS - sets with at least one common element
Example : Let E = { x∣x is the set of positive factors of 4 }
and Let F = { x∣x is the set of positive factors of 9 }
Since E = {1, 2, 4} and F = {1, 3, 9} wherein element 1 can be found both sets E and F,
then these sets are said to be joint sets.

DISJOINT SETS – set have no common element
Example : Let E = { x∣x is the set of even numbers }
and Let F = { x∣x is the set of odd number }
SUBSET – set wherein every element of which can be found on the second set. ⊂
If the first set equals the second set, then it is an improper subset. The symbol ⊆ is used to
mean improper subset.
A null set is always a subset of any given set is considered an improper subset of the given set.
Other than the set itself and the null set, are all considered proper subsets.
The set containing all the subsets of the given set with n number of elements is called the power
sets with number of elements.
Example.
Let T = { x∣x is the first three letters in English alphabet }
= { a, b, c, }
Since set T has three elements, then the power set has = 8 elements.
Let S = { {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, T, Ø }
1.4 OPERATIONS ON SETS
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1. UNION OF SETS A and B (denoted by A U B ) - set whose elements are found in A or B or in
both.
Example. Let A = {a, b, c, d}
B = { c, d, e }
Then A U B = {a, b, c, d, e}
2. INTERSECTION of sets A and B (denoted by A ∩ B ) – set whose elements are common to
both sets.
Example. Let A = {a, b, c, d}
B = { c, d, e }
Then A ∩ B = { c, d}
3. DIFFERENCE of sets A and B (denoted by A - B ) – set whose elements are found in set A but
not in set B.
Example. Let A = {a, b, c, d}
B = { c, d, e }
Then A - B = { a, b }
4. COMPLIMENT OF Set A (denoted by A’) – the set of elements found in the universal set but not
in set A.
Example. Let U = {a, b. c, d, e}
A = {a, b, c, d}
B = { c, d, e }
Then A’ = { e }
B’ = { a, b }
1.5 Venn – Eular Diagrams or simply Venn Diagrams
- Pictorial Presentation of Relationship and Operation of Sets
- Universal Set is usually represented by a rectangle
- While circles within the rectangle usually represent its subset
- Shaded region in the given diagrams illustrates the sets relation or operation.
2. FUNCTION
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

A relation in which input has only one output
Mathematical entities that give unique outputs to particular input
A function (F) form a set of elements X to a set of elements Y is a rule that assigns to each element in
x in X, exactly one element in y in Y.
For Example: Let set X consists of four students and Set Y consists of their favorite subjects, respectively
X = { Alyssa, Elijah, Steph, Shei }
Y = { Chemistry, Math, Physics, Statistics }
Then the result is a set of ordered pairs of the form (x, y), written as:
{(Alyssa, Chemistry), (Elijah, Math), (Steph, Physics),( Shei, Statistics) }
3. RELATIONS
- A set of inputs and outputs often expressed as ordered pairs.
A relation can be represented as a


o
mapping diagram ( mapping or pairing from domain to the range is one way to show
correspondence in a relation)
graph relation ( Cartesian plane)
To determine whether the given ordered pairs is function or not. We will used the VERTICAL LINE
TEST (if all vertical lines intersect th
o
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