Chapter 1.

Section 2: Mathematical
Language and Symbols
Conventions of a Mathematical Language
A Mathematical Convention is a fact, name, notation, or usage which is generally agreed upon by
mathematicians. For instance, the fact that one evaluates multiplication before addition in the
expression.

A mathematical language uses symbols, instead of words, to communicate mathematical ideas.

The syntax and structure can be categorized into 5 forms:
• Numbers: 0,1,2,…(represent quantity: Nouns in the alphabet)
• Operation symbols: +, – , *, ÷ (act as connectives in math sentences)
• Relation symbols: =, ≠, <, ≤, >, ≥ (for comparison, act as verbs)
• Grouping symbols: (), [], {} (to associate groups of numbers and operators)
• Variables: x, y, n, A, V (letters to represent quantities, act as pronouns)
Characteristics of Mathematical
Language
Mathematical Language is the system used to communicate mathematical ideas.
It 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 language is an efficient and powerful tool for mathematical expression, exploration,
reconstruction after exploration, and communication.

It is precise, concise and powerful.

Precise – able to make very fine distinctions (definition)
Concise – able to say things briefly (symbol)
Powerful – able to express complex thoughts with relative ease
Expression vs. Sentences
An expression (mathematical expression) is a finite combination of symbols that is well-defined
according to rules that depend on the context.
- It is the correct arrangement of mathematical symbols to represent the objects of interest,
does not contain a complete thought, and cannot be determined if it’s true or false.

A sentence (mathematical sentence) is a statement about two expressions, either using numbers,
variables or a combination of both.
- It is the correct arrangement of mathematical symbols that states a complete thought,and can
be determined if it’s true or false.
- Presence of the equal (=) symbol.
Expression vs. Sentences
Example:

Expression Sentence
3 2+3 = 6
2+5 10 ÷ 2 = 5
X+3 7 – 3 + 10 = 14
𝑥2 15 ÷ 3 + 2 = 3
𝑥+𝑎 2 15 ÷ (3+2) = 3
Conventions in Mathematics
• Capital Letters are used to name a dot or a set and label vertices of a polygon
• PEMDAS / GEMDAS rule
• Use the name of the definition, postulate or theorem instead of writing its
content (Ex. Using the expression “2k+1” to denote an odd number)
• Always express fractions in simplest form
• Distances in a coordinate system are non-negative
• When constructing a Venn diagram, we represent universal set using a
rectangle and various sets of objects using circular regions including those
regions formed by intersection of circles
Four Basic Concepts

• Sets
• Functions
• Relations
• Binary Operations
Sets

Set Theory is the branch of mathematics that studies sets or

the mathematical science of the infinite
George Cantor (1845-1918) – a German Mathematician
considered to be the founder of set theory as a mathematical
discipline.
A set is a well-defined collection of objects.
Sets

In Mathematics, a set is a well-defined collection of distinct

objects called elements/members of the set.
Examples:
a. The set of counting numbers from 1 to 10.
b. The set of English Alphabets.
c. The set of integers.
The Language of Sets

What is an element of a set?

• Pertains to each member of the set
• The notation ∈ denotes that an object is an element of the set
Example: A = {1,2,3,4,5,6,7,8,9,10}
1∈A 5∈A 19 ∉ A
Note: ∉ - not an element of
Methods of Describing Sets

Roster or Tabular Method Rule Method or Set Builder Notation

- It is done by listing or tabulating the
elements of the set - It is done by stating or describing the
common characteristics of the elements of
the set. We use the notation D = {x|x …}
Example:
A = {2,4,6,8,10,12,14,16,18,20} Example:
B = {January, February, March, April, May, D : {x|x ∈ R}
June, July, August, September, October,
November, December} E : {y|y is a multiple of 3 and is less than 25}
C = {Andrews, Carig, Lallo, Gonzaga, Appari,
Lasam, Piat, Sanchez Mira, Solana}
Special Sets
Standard notations use
Definition
to define some sets

N Set of all natural numbers

Z Set of all integers

Set of all rational

Q
numbers

R Set of all real numbers

Set of all complex

C
numbers
Terminologies on Sets
1. Unit Set
- a set that contains only one element
2. Empty Set or Null Set ({ } or ∅)
- a set that has no element
3. Finite Set
- a set with ‘countable’ elements
4. Infinite Set
- a set which elements are not countable or has no end
5. Cardinal Number/Set Cardinality (n)
- total number of elements in a set
Terminologies on Sets

6. Equal Set
- Two sets are said to be equal if they have equal cardinality and identical
elements
7. Equivalent Set
- Two sets are considered equivalent sets if they have the same cardinality
8. Universal Set (U)
- universal set is the set of all elements under discussion
9. Joint Sets
- sets are said to be joint sets if they have at least one common element
10. Disjoint Sets
- are mutually exclusive sets or sets that have no common element
Subsets
Set A is said to be a subset of Set B if all the elements of Set A
are also present in Set B. In other words, set A is contained
inside Set B.
Example: If set A has {X, Y} and set B has {X, Y, Z}, then A is the
subset of B because elements of A are also present in set B.
A⊆B
Note: The number of subsets of a certain set can be identified using the
formula 2n, where n is the set cardinality.
Proper Subset (⊂)
Set C is considered to be a proper subset of Set D if Set D
contains at least one element that is not present in Set C.
Example: If set C has elements {12, 24} and set D has
elements {12, 24, 36}, then set C is the proper subset of D
because 36 is not present in the set C.
C⊂D
Read as Set C is a ‘proper subset’ of set D
Improper Subset (⊆)
Set P is considered to be an improper subset of Set Q if set P
contains all the elements of set Q.
Example: If set P has elements {2,4,6,8,10} and set Q has
elements {10,8,6,4,2}, then set P is the improper subset of Q.
P⊆Q
Read as set P is the ‘improper subset’ of set Q
Power Set
The Power Set of A is the set whose elements are the subsets
of A. It is denoted by p(A) and the cardinal number of the
power set is n(p(A)) = 2n(A).

Example: Given the set M = {5,10,15}, the subsets are

{ }, {5}, {10}, {15}, {5,10}, {5,15}, {10,15},{5,10,15} and the power set is
p(A) = {{ }, {5}, {10}, {15}, {5,10}, {5,15}, {10,15},{5,10,15}}
Operations on
Sets
Union of Sets
• The union of sets A and B, denoted by A∪B
A∪B = {x|x ∈ A or x ∈ B}

Ex. If A = {1,2,3} and B = {4,5}, then A∪B = {1,2,3,4,5}

If A = {1,2,3} and B = { 1,3,5,7}, then A∪B = {1,2,3,5,7}
Illustration A∪B:
Intersection of Sets
• The intersection of sets A and B, denoted by A∩B
A∩B = {x|x ∈ A and x ∈ B}
Ex. If A = {1,2,3} and B = {1,2,4,5}, then A∩B = {1,2}

If A = {1,2,3} and B = { 4,5}, then A∪B = { } or ∅

Illustration A∩B:
Difference of Sets
• The intersection of sets A and B, denoted by A – B
A – B = {x|x ∈ A and x ∉ B}
Ex. If A = {1,2,3} and B = {1,2,4,5}, then A – B = {3}

If A = {1,2,3} and B = { 4,5}, then A – B = {1,2,3}

Illustration A – B:
Complement of Sets
• For set A, the difference U – A, where U is the universal set is called
the complement of a Denoted by 𝑨𝒄 or 𝑨’. Thus it is the set of
everything that is not in A.
Ex. Let U = {a,e,i,o,u} and A = {a,e}, then 𝑨𝒄 = {i,o,u}

Illustration A – B:
Cartesian Product
• Given sets A and B, the cartesian product of A and B denoted by A x B
and read as “A cross B”, is a set of all ordered pair (a,b) where a is in A
and b is in B.
A x B = {(a,b) | a ∈ A and b ∈ B}
Note: A x B is not equal to B x A
Ex. If A = {1,2,3} and B = {a,b,c}, then
A x B = {(1,a),(1,b),(1,c),(2,a),(2,b),(2,c),(3,a),(3,b),(3,c)}
What is B x A?
B x A = {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3),(c,1),(c,2)(c,3)}
Symmetric Difference
• The symmetric difference of to sets A and B (A Δ B) is the set of
elements in A ∪ B that are not in A ∩ B.
A Δ B = {x|x ∈ A ∪ B and x ∉ A ∩ B}

Example: Let A = {a, e, i, o, u} and B = {a, b, c, d, e}

A ∪ B = {a, b, c, d, e, i, o, u} and A ∩ B = {a, e}, thus,

A Δ B = {b, c, d, i, o, u}
Problem Set
A. Identify whether the sets are joint or disjoint sets.
1. A is the set of real numbers
B is the set of Integers
2. C = {3,6,9,12,15,18,21,24,27,30}
D is the set of even numbers
3. E = {a,b,c,d,e,f}
F = {v,w,x,y,z}
4. G={a,e,i,o,u}
H={e,f,h,g,i,j}
5. I = {2,4,6,8,10,12,14,16,18,20}
J is the set of prime numbers
Problem Set
B. Solve for the following
1. Let A = {1,2,3,4,5,6,7,8,9,10} and B = {2,4,6,8,10}. Solve for:
a. A-B
b. B-A
2. If the set U is the set of the English Alphabet and set C
is the set of the letters in my first name, what is 𝑪𝒄 ?
3. Let D = {x,y,z} and E = {1,2,3,4,5}.Solve for:
a. DxE
b. ExD
4. Let S = {a,b,c,d,e} and T = {a,e,i,o,u}. Find S Δ T.