College of Education
STUDY GUIDE
GE 3: MATHEMATICS IN THE MODERN WORLD
Mathematical Language and Symbols
The mathematical language is the system used to communicate mathematical
ideas. This language consists of some natural language using technical terms
(mathematical terms) and graphical conventions that are uncommon to mathematical
discourse, supplemented by a highly specialized symbolic notation for mathematical
formulas. The mathematical notation used for formulas has its own grammar and shared
by mathematicians anywhere in the globe.
At the end of the lesson, you should be able to:
• Discuss the language, symbols and conventions of mathematics
• Explain the nature of mathematics as a language
• Perform operations on mathematical expression correctly, its basic concepts and
logic
• Appreciate that mathematics is a useful language.
Key Concepts:
✔ Language of math
✔ Mathematical expression
✔ Mathematical sentence
✔ Mathematical convention
✔ BODMAS / PEMDAS
✔ Four basic concepts of math
✔ Logic
This material is exclusively for Northeastern College students ONLY. Any redistribution or reproduction of part or
of all its contents is prohibited. © 2023
This material is exclusively for Northeastern College students ONLY. Any redistribution or reproduction of part or
of all its contents is prohibited. © 2023
Mathematical Expression versus Mathematical Sentence
• Expression
✓ A finite combination of symbol that is well-defined according to the rules that
depend on the context.
✓ A correct arrangement of mathematical symbols used to represent a
mathematical object of interest.
✓ Does not state a complete thought.
✓ Does not make sense to ask if an expression is true or false
• Sentence
✓ A correct arrangement of mathematical symbols that states a complete thought
✓ Can always be true, always false, sometimes true or sometimes false
Conventions in the Mathematical Language
• Mathematical convention
✓ fact, name, notations or usage which generally agreed upon by the
mathematicians
✓ = (equal), < (less than), > (greater than), + (addition), - (subtraction), x
(multiplication), ÷ (division), ∈ (element), ∀ (for all), ∃ (there exist), ∞ (infinity),
→ (implies), ↔ (if and only if), ≈ (approximately), ∴ (therefore)
✓ Greek letters (α, β, γ, . . .), Latin/English letters (a, b, c, . . .; A, B, C, . . .),
numbers (1, 2, 3, . . .), superscripts (xn), subscripts (xn)
This material is exclusively for Northeastern College students ONLY. Any redistribution or reproduction of part or
of all its contents is prohibited. © 2023
Evaluate
(11 – 5) x 2 – 3 + 1
Evaluate
10 / 2 + 12 / 2 x 3
Simplify
4 – 3[4 – 2(6 – 3)] / 2
Simplify
16 – 3(8 – 3)2 / 5
Language of Sets
▪ Set theory – branch of mathematics that studies sets or the mathematical science
of the infinite
▪ Georg Cantor
▪ Set – well-defined collection of objects
– Elements or members – objects
– ∈ - an object is an element of a set
– ∉ - an object is not an element of a set
Two ways to represent sets: (1) roster method, and (2) rule method
▪ Roster method – elements of the set are enumerated or separated by a comma
(tabulation method)
– E = {a, e, i, o, u}
▪ Rule method – used to describe the elements or members of the set (set-builder
notation)
– E = {x|x is a collection of vowel letters}
Some Terms in Set
– Finite Set
▪ Limited
▪ Countable
▪ A = {x|x is a 5 kg salt}
– Infinite Set
▪ Unlimited
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▪ Uncountable
▪ A = {strands of hair}
– Unit Set
▪ One element only
▪ Singleton
▪ A = {I}
– Empty Set
▪ No element
▪ Null set
▪ ⌀ or {}
– Universal Set
▪ It has all of the elements
– Cardinal Number
▪ Cardinality
▪ Number of elements
▪ n(A)
▪ E = {a, e, i, o, u}, ∴ n(E)=5
Kinds of Set
– Subset
▪ ⊆
▪ A ⊆ B, iff every ∈ A is in ∈ B
▪ A = {a, e, i, o, u}, B = {a, e, i, o, u, v, w}, ∴ A ⊆ B
– Proper Subset
▪ ⊂
▪ A ⊂ B, iff every ∈ A is in B but there is at least one ∈ B that is not in
A
▪ A = {a, e, i, o, u}, B = {a, e, i, o, u, v}, ∴ A ⊂ B
– Equal Set
▪ =
▪ A=B
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of all its contents is prohibited. © 2023
▪ A = {a, e, i, o, u}, B = {a, e, i, o, u}, ∴ A = B
– Power Set
▪ ℘ (s) = 2n, n = number of elements
▪ Collection (or sets) of all subset of s
▪ A = {a, e}, ℘(A) = 22=4, ∴ A = {{a}, {e}, {a, e}, ⌀}
Operations on Set
– Union
▪ ⋃
▪ A⋃B
▪ A ⋃ B = {x|x ∈ A ˅ x ∈ B}
▪ A = {1, 2} and B = {3, 4}, ∴ A ⋃ B = {1, 2, 3, 4}
– Intersection
▪ ⋂
▪ A⋂B
▪ A ⋂ B = {x|x ∈ A ˄ x ∈ B}
▪ A = {1, 2, 3} and B = {3, 4}, ∴ A ⋂ B = {3}
– Complement or Absolute Complement
▪ A’
▪ A’ = {x ∈ U | x ∉ A}
▪ U = {a, b, c, d, e, f, g}, A = {a, b, c, d, e}, ∴ A’ = {f, g}
– Difference or Relative Complement
▪ ∼
▪ A ∼ B = {x|x ∈ A ˄ x ∉ B} = A ⋂ B’
▪ A = {a, b, c}, B = {c, d, e}, U = {a, b, c, d, e, f, g} ∴ A ∼ B= {a, b}
– Symmetric Difference
▪ ⊕
▪ A ⊕ B = {x|x ∈ (A⋃B) ˄ x ∉ (A⋂B)} = (A⋃B) ⋂ (A⋂B)’ or (A⋂B) ∼
(A⋂B)
▪ A = {a, b, c}, B = {c, d, e}, U = {a, b, c, d, e, f, g} ∴ A ⊕ B= {a, b, d, e}
– Disjoint or Non-intersecting
▪ ⇔
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of all its contents is prohibited. © 2023
▪ No elements are common
– Ordered Pair (a, b)
▪ a – first component
▪ b – second component
▪ (a, b) ≠(b, a)
– Cartesian Product
▪ = A x B = {(a, b) | a ∈ A and b ∈ B}
▪ A = {a, b, c}, B = {1, 2}, ∴ A x B = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1),
(c, 2)}
Reflection and Research:
✔ Describe the language of mathematics.
✔ What are the characteristics of language of mathematics?
✔ Differentiate mathematical expression from mathematical sentence.
✔ Why do we need to abide the mathematical conventions?
✔ How do you perform operations on mathematical expressions correctly?
✔ Explain the language of sets.
SUMARRY:
✓ Language of mathematics must be precise, concise, and powerful.
✓ Expressions does not state a complete thought.
✓ Mathematical sentence can always be true, always false, sometimes true or
sometimes false
✓ Always remember to solve from left to right in the rule of PEMDAS/BODMAS
✓ Some sets may be finite, infinite, unit, empty, universal and a cardinal number
✓ Kinds of set are subset, proper subset, equal set and power set
✓ Operations on set are union, intersection, complement, difference, symmetric
difference, disjoint, ordered pair and cartesian product
This material is exclusively for Northeastern College students ONLY. Any redistribution or reproduction of part or
of all its contents is prohibited. © 2023
References:
Daligdig, Romeo M. (2019) Mathematics in the Modern Works. Manila, Philippines:
Lorimar Publishing, INC.
Manlulu, Esmeralda A. And Liza Marie M. Hipolito (2019) Course Module for
Mathematics in the Modern. Manila, Philippines: Rex Book Store
Mathematics in the Modern World (2018) Cengage Philippine Edition. Sampaloc,
Manila, Philippines: Rex Book Store, Inc.
Sirug, Winston (2018). Mathematics in the Modern World. CHED Curriculum
Compliant. Manila, Philippines: Mindshapers Co., Inc.
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Prepared By: Checked/Approved By:
Rina A. Guingab, LPT Saranay I. Doyaoen, MS-Math, CPA
Instructor Dean, College of Education
This material is exclusively for Northeastern College students ONLY. Any redistribution or reproduction of part or
of all its contents is prohibited. © 2023
