1) A variable is a symbol, usually a letter, that represents a value that may change. Variables can be classified based on their relationship (independent or dependent) or type (continuous or discrete).
2) Set theory deals with collections of distinct objects called sets. A set can be defined, represented with capital letters, and contain elements enclosed in braces. Basic set operations include union, intersection, and difference.
3) There are different types of sets such as empty, finite, infinite, proper and improper subsets, disjoint and joint sets, equivalent and equal sets. Understanding sets and their properties is fundamental to mathematics.
2. Variable
is a symbol, usually a letter that holds a value that may
increase or decrease over time or takes different values in
different situation.
Example:
𝑥2 − 5𝑥 + 6
In this polynomial, variable x is used which may have different
values. If we will try to substitute a value of 3 to variable x, the
whole polynomial will have an equivalent value of 0.
3. For every case, the value of the variable may be different, or sometimes
might not exist. Sometimes a variable is represented using a symbol
like for example in trigonometry. An angle is often represented using a
symbol called theta (θ). The variables in an expression or equation may
be categorized into different types, and is listed below:
According to Functional Relationship
• Independent Variable-called the predictor variable
• Dependent Variable - called the criterion variable
Examples:
The academic performance of students in Mathematics (y)
depends on their study habits and their attitudes towards the subject
(x).
Independent Variable - Student's study habits and attitudes(x)
Dependent Variable-academic performance of students in Mathematics
(y)
4. According to Continuity of Values
• Continuous Variables-variables that can be expressed in
decimals
Examples: Price of commodities, grades, height
• Discrete or Discontinuous Variables-variables that can't be
expressed in decimals
Examples: number of students, cars, houses
5. Knowing the classification of the variables used in an equation and
expression is important so that you will know if let's say a decimal
answer is accepted or not to represent the said variable. Like let's say
we are solving a problem in trigonometry and we are trying to identify
the third angle of a triangle as shown in figure below. Since,we don't
know yet the value of the angle at vertex A, we can represent its value
as θ and use this symbol in creating an equation.
Then, solving for the angle at vertex A, we can say:
θ+30+70=180 [using the principle that sum of angles in triangle is
180 degrees]
θ =80 degrees
6. Sets
Set theory is a topic in Mathematics that deals with the logic involving group of objects that
are most of the time have commonalities among them. A set to be studied does not necessarily mean to
be consisting of numbers only but can also be group of colors, country, names, etc. Set theory is very
significant in understanding all branches of mathematics. It is considered as the basis of all the other
mathematics.
-Set
a well-defined collection of distinct objects
b. set can be denoted by a capital letter
Example: A={Even numbers}, Q = {Primary colours}, D = {0, 2,5}
-Element
a. The object that make up a set
b. Enclosed by braces and separated by commas
Example: {1, 2,3,4,5}, {2,4,6,...}, {red, blue, yellow]
- Universal Set
a. A set containing all the existing elements
b. All the sets are subset of the universal set
C. Usually denoted as U
-Order of the set
a. Describe the number of elements inside a given set
Example: {1,21,31,41,51,61} Order=6
{pants, shorts, trousers} Order=3
7. Types of Sets:
Sets are categorized into different types depending on their property and
characteristics. Listed below are some of the fundamental types of sets.
1. Empty set
a. A set that has no element in it
b. Also called as null set
2. Finite Set
a. Consist of countable numbers of elements
b. Has a determinate number of elements
Example: {2,4,6,8,10}
3. Infinite Set
a. Has indeterminate number of elements
Example: {even numbers}
4. Proper Subset of a Set
a. A set whose elements are found in a larger set
b. Part of a larger set
Example: Set = {Odd numbers} Subset= {1,3,11}
5. Improper Subset of a Set
a. A set is an improper subset of a given set if they are equal sets
b. Null set is an improper subset of any set
8. 6.Disjoint Sets
a. Two or more sets with no common elements
Example: A={vowels} and B = {consonants}
7.Joint Sets
a. Two or more sets with at least one common elements
Example: A ={composite numbers} and B ={odd numbers}
8.Equivalent Sets
a. Two or more sets with same number of elements
Example: {1,2,3,4,5} and {6,7,8,9,10}
9.Equal Sets
a. Two or more sets with the same elements
Example: {red, blue, yellow} and {primary colors}
9. Basic Operations in Sets:
UNION OF SETS
Union of two sets is simply the elements resulted from
combining the two sets.
Union is denoted by U. (i.e. union of set A and B is written
as A U B).
10. INTERSECTION OF SETS
Intersection of two sets is simply the elements that are
common to the two sets
Intersection is denoted by the symbol
11. DIFEERENCE OF SETS
The difference of two sets is the elements resulted when
the elements common two the two sets are subtracted from the
minuend set.
12. COMPLEMENT(U')
Complement of sets is the elements found in the universal set that is not
found in any of the subsets of the universal set.
The other way of saying this is that U' = U - (A U B)
Example:
A={odd numbers less than 15} = {1, 3, 5, 7,9, 11, 13}
B={prime numbers less than 15} = {2,3,5,7,11,13}
U = {numbers less than 1 15} = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
U'={4,6,8,10,12,12,14}
