Mathematical Language & Symbols
Language is the system of words, signs and symbols which people use to express ideas, thoughts and feelings.
Mathematical Language is the system used to communicate mathematical ideas.
Four main Actions attributed to Problem Solving and Reasoning
- Modeling and Formulating
- Transforming and Manipulating
- Inferring
- Communicating
Characteristics of Mathematical Language
- Mathematics is about ideas - relationship, quantities, processes, measurements, reasoning and etcetera.
- According to Jamison(2000) the use of language in Mathematics differ from the language of the ordinary speech in three important ways.
- Mathematical Language is non-temporal. There is no past, present, and future in mathematics.
- Mathematical Language is devoid of emotional content.
- Mathematical Language is precise.
Advantage of Mathematical Notation
Symbolic and Graphical is that it is highly compact and focused
Mathematical Expressions
- Mathematical expression consists of terms.
- The term of a mathematical expression contains a number and a letter separated by at least one of the fundamental operations.
- In algebra, variables or letters are used to represent unknown quantities.
Mathematical expressions may be classified according to the number of terms as follows:
- Monomial contains one term only.
Examples are: 2x; 5y; 8z; 4m
- Binomial contains two terms.
Examples are: 2x+5y; 5y-9z; 8z+10m
- Trinomial contains three terms.
Examples are: 2x+5y-3; 4m+n-10
- Multinomial contains four or more terms.
Examples are: 2a-6b+7c-9; 4x+18y+2z+10
Mathematical Sentence
It is a combination of two mathematical expressions using a comparison operator.
Types of Mathematical Sentence
- Open Sentence - it is not known whether or not the mathematical sentence is true or false.
ex. 2xy < 3y
- Closed Sentence - the mathematical sentence is known to be either true or false.
ex. 2(x + y) = 2x + 2y
Conventions of Mathematical Language
Context refers to the particular topics being studied. It is important to understand the context to understand mathematical symbols.
Convention is a technique used by mathematicians, engineers, scientist in which each particular symbol has particular meaning.
Four basic concepts:
- Set is a well-defined collection of distinct objects.
- Functions are mathematical quantities that give unique outputs to particular inputs.
- Relations are correspondence between a first set of variables such that for some elements of the first set variables, there correspond at least two elements of the second set of variables.
- Binary Operations are rules for combining two values to produce a new value.
Sets
Two ways to describe a set
- Roster or Tabular Method
ex. E = {a, e, i, o, u}
- Rule or Descriptive Method
ex. E = {x|x is a collection of vowels in the English Alphabet}
Kinds of Sets
- Empty/Null/Void Set is a set that contains no element.
ex. C = {x|x is an integer less than 2 but greater than 1}
- Finite Set is a set that contains a countable number of elements.
ex. A = {a, b, c}
- Infinite Set is a set that contains an uncountable number of elements.
ex. A = {…, -2, -1, 0, 1, 2, …} and B = {x|x is a set of whole numbers}
- Universal Set is a set that contains all elements under consideration.
ex. U = {x|x is an animal in Manila Zoo} and U = {1, 2, 3, …,100}
- Unit Set (Singleton) is a set that contains only one element.
ex. E = {x|x is a whole number greater than 1 but less than 3}
- Equal Sets are sets that contain exactly the same elements.
ex. A = {0, 1, 2, 3} B = { 2, 0, 1,3}
- Equivalent Sets are sets that contain the same number of elements.
ex. A = {a, b, c, d} B = { 2, 0, 1, 3}
- Joint Sets are sets that have at least one common element.
ex. A = {a, b, c, d} B = {d, e, f, g, h}
- Disjoint Sets are sets that contain no common element.
ex. A = {a, b, c, d} B = { e, f, g, h}
- Venn Diagram and Set Operations
- Similar to numbers, we can perform certain mathematical operations on sets.
- The principal operations are the intersection, union, difference, symmetric difference, and the complement of sets.
- Venn diagrams are used to visualize set operations.
- In a Venn diagram, a rectangle shows the universal set, and all other sets are usually represented by circles within the rectangle. The shaded region represents the result of the operation.
Operations on Sets
Unions
Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B.
Examples:
{1, 2} ∪ {1, 2} = {1, 2}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
Some basic properties of unions:
A ∪ B = B ∪ A.
A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ∪ A = A
Intersections
A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.
Examples:
{1, 2} ∩ {1, 2} = {1, 2}.
{1, 2} ∩ {2, 3} = {2}.
{1, 2} ∩ {3, 4} = ∅.
Some basic properties of intersections:
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ A = A.
A ∩ ∅ = ∅.
Complements
Two sets can also be "subtracted". The relative complement of B in A (also called the set theoretic difference of A and B), denoted by A \ B (or A − B), or A’ or A^c , the set of all elements that are members of A, but not members of B.
Examples:
{1, 2} \ {1, 2} = ∅.
{1, 2, 3, 4} \ {1, 3} = {2, 4}.
Some basic properties of complements include the following:
A \ B ≠ B \ A for A ≠ B.
A ∪ A′ = U.
A ∩ A′ = ∅.
(A′)′ = A.
The Venn Diagram
A Venn diagram, also called primary diagram, set diagram or logic diagram, is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set.
Set of operations on Venn Diagram
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