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MATHEMATICS-1A STUDY MATERIAL

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1st Year Maths IA Study Material

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German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description.

Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. In this chapter, we will cover the different aspects of Set Theory.

Set - Definition

A set is an unordered collection of different elements(a set is a Well define Collection of objects). A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

Some Example of Sets

A set of all positive integers
A set of all the planets in the solar system
A set of all the days in a week
A set of all the states in India
A set of all the lowercase letters of the alphabet

Representation of a Set

Sets can be represented in two ways −

Roster or Tabular Form
Set Builder Notation

Roster or Tabular Form

The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.

Example 1 − Set of vowels in English alphabet,

Example 2 − Set of odd numbers less than 10,

Set Builder Notation

The set is defined by specifying a property that elements of the set have in common. The set is described as

Example 1 − The set is written as −

Example 2 − The set is written as −

If an element x is a member of any set S, it is denoted by and if an element y is not a member of set S, it is denoted by .

Example − If but

Some Important Sets

N − the set of all natural numbers =

Z − the set of all integers =

Z⁺ − the set of all positive integers

Q − the set of all rational numbers

R − the set of all real numbers

W − the set of all whole numbers

Cardinality of a Set

Cardinality of a set S, denoted by , is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is .

Example −

If there are two sets X and Y,

denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.
denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y.
denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective.
and then . The sets X and Y are commonly referred as equivalent sets.

Types of Sets

Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper, singleton set, etc.

Finite Set

A set which contains a definite number of elements is called a finite set.

Example − and

Infinite Set

A set which contains infinite number of elements is called an infinite set.

Example − and

Subset

A set X is a subset of set Y (Written as ) if every element of X is an element of set Y.

Example 1 − Let, and . Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write .

Example 2 − Let, and . Here set Y is a subset (Not a proper subset) of set X as all the elements of set Y is in set X. Hence, we can write .

Proper Subset

The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of set Y (Written as ) if every element of X is an element of set Y and .

Example − Let, and . Here set since all elements in are contained in too and has at least one element is more than set .

Universal Set

It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as .

Example − We may define as the set of all animals on earth. In this case, set of all mammals is a subset of , set of all fishes is a subset of , set of all insects is a subset of , and so on.

Empty Set or Null Set

An empty set contains no elements. It is denoted by . As the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.

Example − and

Singleton Set or Unit Set

Singleton set or unit set contains only one element. A singleton set is denoted by .

Example − =

Equal Set

If two sets contain the same elements they are said to be equal.

Example − If and , they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

Equivalent Set

If the cardinalities of two sets are same, they are called equivalent sets.

Example − If and , they are equivalent as cardinality of A is equal to the cardinality of B. i.e.

Overlapping Set

Two sets that have at least one common element are called overlapping sets.

In case of overlapping sets −

Example − Let, and . There is a common element ‘6’, hence these sets are overlapping sets.

Disjoint Set

Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, disjoint sets have the following properties −

Example − Let, and , there is not a single common element, hence these sets are overlapping sets.

Venn Diagrams

Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets.

Examples

Venn Diagram

Set Operations

Set Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.

Set Union

The union of sets A and B (denoted by ) is the set of elements which are in A, in B, or in both A and B. Hence, .

Example − If and B = , then . (The common element occurs only once)

Set Union

Set Intersection

The intersection of sets A and B (denoted by ) is the set of elements which are in both A and B. Hence, .

Example − If and , then .

Set Intersection

Set Difference/ Relative Complement

The set difference of sets A and B (denoted by ) is the set of elements which are only in A but not in B. Hence, .

Example − If and , then and . Here, we can see

Set Difference

Complement of a Set

The complement of a set A (denoted by ) is the set of elements which are not in set A. Hence, .

More specifically, where is a universal set which contains all objects.

Example − If then

Complement Set

Cartesian Product / Cross Product

The Cartesian product of n number of sets denoted as can be defined as all possible ordered pairs where

Example − If we take two sets and ,

The Cartesian product of A and B is written as −

The Cartesian product of B and A is written as −

Power Set

Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is . Power set is denoted as .

Example −

For a set let us calculate the subsets −

Subsets with 0 elements − (the empty set)
Subsets with 1 element −
Subsets with 2 elements −
Subsets with 3 elements −
Subsets with 4 elements −

Hence,

Note − The power set of an empty set is also an empty set.

Partitioning of a Set

Partition of a set, say S, is a collection of n disjoint subsets, say that satisfies the following three conditions −

does not contain the empty set.
The union of the subsets must equal the entire original set.
The intersection of any two distinct sets is empty.

Example

Let

One probable partitioning is

Another probable partitioning is

Bell Numbers

Bell numbers give the count of the number of ways to partition a set. They are denoted by where n is the cardinality of the set.

Example −

Let ,

The alternate partitions are −

Hence

Whenever sets are being discussed, the relationship between the elements of the sets is the next thing that comes up. Relations may exist between objects of the same set or between objects of two or more sets.

Definition and Properties

A binary relation R from set x to y (written as $x R y$ or $R (x, y)$ ) is a subset of the Cartesian product $x \times y$ . If the ordered pair of G is reversed, the relation also changes.

Generally an n-ary relation R between sets $A_{1}, \dots, a n d A_{n}$ is a subset of the n-ary product $A_{1} \times \dots \times A_{n}$ . The minimum cardinality of a relation R is Zero and maximum is $n^{2}$ in this case.

A binary relation R on a single set A is a subset of $A \times A$ .

For two distinct sets, A and B, having cardinalities m and n respectively, the maximum cardinality of a relation R from A to B is mn.

Domain and Range

If there are two sets A and B, and relation R have order pair (x, y), then −

The domain of R, Dom(R), is the set ${x | (x, y) \in R f o r s o m e y i n B}$
The range of R, Ran(R), is the set ${y | (x, y) \in R f o r s o m e x i n A}$

Examples

Let, $A = {1, 2, 9}$ and $B = {1, 3, 7}$

Case 1 − If relation R is 'equal to' then $R = {(1, 1), (3, 3)}$ Dom(R) = ${1, 3}, R a n (R) = {1, 3}$
Case 2 − If relation R is 'less than' then $R = {(1, 3), (1, 7), (2, 3), (2, 7)}$ Dom(R) = ${1, 2}, R a n (R) = {3, 7}$
Case 3 − If relation R is 'greater than' then $R = {(2, 1), (9, 1), (9, 3), (9, 7)}$ Dom(R) = ${2, 9}, R a n (R) = {1, 3, 7}$

Representation of Relations using Graph

A relation can be represented using a directed graph.

The number of vertices in the graph is equal to the number of elements in the set from which the relation has been defined. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. If there is an ordered pair (x, x), there will be self- loop on vertex ‘x’.

Suppose, there is a relation $R = {(1, 1), (1, 2), (3, 2)}$ on set $S = {1, 2, 3}$ , it can be represented by the following graph −

Relation

Types of Relations

The Empty Relation between sets X and Y, or on E, is the empty set $\emptyset$
The Full Relation between sets X and Y is the set $X \times Y$
The Identity Relation on set X is the set ${(x, x) | x \in X}$
The Inverse Relation R' of a relation R is defined as − $R^{'} = {(b, a) | (a, b) \in R}$ Example − If $R = {(1, 2), (2, 3)}$ then $R^{'}$ will be ${(2, 1), (3, 2)}$
A relation R on set A is called Reflexive if $\forall a \in A$ is related to a (aRa holds)Example − The relation $R = {(a, a), (b, b)}$ on set $X = {a, b}$ is reflexive.
A relation R on set A is called Irreflexive if no $a \in A$ is related to a (aRa does not hold).Example − The relation $R = {(a, b), (b, a)}$ on set $X = {a, b}$ is irreflexive.
A relation R on set A is called Symmetric if $x R y$ implies $y R x$ , $\forall x \in A$ and $\forall y \in A$ .Example − The relation $R = {(1, 2), (2, 1), (3, 2), (2, 3)}$ on set $A = {1, 2, 3}$ is symmetric.
A relation R on set A is called Anti-Symmetric if $x R y$ and $y R x$ implies $x = y \forall x \in A$ and $\forall y \in A$ .Example − The relation $R = {(x, y) \to N | x \leq y}$ is anti-symmetric since $x \leq y$ and $y \leq x$ implies $x = y$ .
A relation R on set A is called Transitive if $x R y$ and $y R z$ implies $x R z, \forall x, y, z \in A$ .Example − The relation $R = {(1, 2), (2, 3), (1, 3)}$ on set $A = {1, 2, 3}$ is transitive.
A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive.Example − The relation $R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}$ on set $A = {1, 2, 3}$ is an equivalence relation since it is reflexive, symmetric, and transitive.A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. The third and final chapter of this part highlights the important aspects of functions.

Function - Definition
An algebraic function is a type of equation that uses mathematical operations. An equation is a function if there is a one-to-one relationship between its x-values and y-values.

Algebraic Functions

An algebraic function is a function that involves only algebraic operations, like, addition, subtraction, multiplication, and division, as well as fractional or rational exponents. Think of an algebraic function as a machine, where real numbers go in, mathematical operations occur, and other numbers come out.

We call the numbers going into an algebraic function the input, x, or the domain. Any number can go into a function as long as it is not divided by zero or does not produce a negative square root. A function can preform many mathematical operations with a domain as long as the range is one value for each domain used. We call the numbers coming out of a function the output, y, or the range. Remember, one value in, one value out.

There are many different types of algebraic functions: linear, quadratic, cubic, polynomial, rational, and radical equations. In this next part of the lesson, we'll learn about a couple of different methods we can use to identify them.

Function Machine

Tables

One way of identifying an algebraic function is through the use of a table, which can show us if there is one domain and one range. Sometimes functions add to the domain to get the range, like x + 2. Sometimes functions multiply the domain to get the range, like 3x. Functions may also subtract or divide the domain or use a combination of operations to produce the range. As long as the rule of 'one in/one out' is kept in place, the function exists.

If an algebraic function says to add two to the domain, we can create a table to show the function:

As you can see, for every domain, we have one range. These pairs of x values- and y-values are called ordered pairs because we put them in order (x,y).

We can also turn our table into ordered pairs to show a function: (1,3), (4,6), (-2,0) and (-3,-1) where there is one x-value for every one y-value.

Graphs

We can also use graphs to identify functions by plotting ordered pairs onto a Cartesian Coordinate System, where the x-values are on the horizontal line and the y-values are on the vertical line. Where the ordered pairs meet is where the point is graphed. If we plot the points, we end up with a straight line, so the function, x + 2, is considered a linear function and can be written in functional notation as f(x) = x + 2. The f(x) is just another way to write y, which we call the f-function. It is a way for us to identify the different functions, instead of calling them all y = ...

Linear Function

Vertical Line Test

We know a graph is a function if it can pass the vertical line test. In this test, if we place a vertical line anywhere on a graph, it will cross in only one place. If a vertical line crosses in two places on a graph, it is in conflict with the one in, one out rule. So, it is not a function.

Vertical Line Test on Linear Function

Here is an example of a graph that is not a function.

Not a Function

Examples of Functions

As we said at the beginning of the lesson, there are many types of functions, such as the quadratic function and the cubic function. Let's start with a quadratic function.

The quadratic function: g(x) = x^2 - 3. First, let's create a table.

The domain can be any real number, this is why the x-value or domain is called the independent variable. Here we'll use -2, -1, 0, 1, and 2 for the domain.

The range or y-value is called the dependent variable because it depends on what we use for the x-term.

A function or mapping (Defined as $f : X \to Y$ ) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.

Function ‘f’ is a relation on X and Y such that for each $x \in X$ , there exists a unique $y \in Y$ such that $(x, y) \in R$ . ‘x’ is called pre-image and ‘y’ is called image of function f.

A function can be one to one or many to one but not one to many.

Injective / One-to-one function

A function $f : A \to B$ is injective or one-to-one function if for every $b \in B$ , there exists at most one $a \in A$ such that $f (s) = t$ .

This means a function f is injective if $a_{1} \neq a_{2}$ implies $f (a 1) \neq f (a 2)$ .

Example
- $f : N \to N, f (x) = 5 x$ is injective.
- $f : N \to N, f (x) = x^{2}$ is injective.
- $f : R \to R, f (x) = x^{2}$ is not injective as $(- x)^{2} = x^{2}$
Surjective / Onto function

A function $f : A \to B$ is surjective (onto) if the image of f equals its range. Equivalently, for every $b \in B$ , there exists some $a \in A$ such that $f (a) = b$ . This means that for any y in B, there exists some x in A such that $y = f (x)$ .

Example
- $f : N \to N, f (x) = x + 2$ is surjective.
- $f : R \to R, f (x) = x^{2}$ is not surjective since we cannot find a real number whose square is negative.
Bijective / One-to-one Correspondent

A function $f : A \to B$ is bijective or one-to-one correspondent if and only if f is both injective and surjective.

Problem

Prove that a function $f : R \to R$ defined by $f (x) = 2 x - 3$ is a bijective function.

Explanation − We have to prove this function is both injective and surjective.

If $f (x_{1}) = f (x_{2})$ , then $2 x_{1} - 3 = 2 x_{2} - 3$ and it implies that $x_{1} = x_{2}$ .

Hence, f is injective.

Here, $2 x - 3 = y$

So, $x = (y + 5) / 3$ which belongs to R and $f (x) = y$ .

Hence, f is surjective.

Since f is both surjective and injective, we can say f is bijective.

Inverse of a Function

The inverse of a one-to-one corresponding function $f : A \to B$ , is the function $g : B \to A$ , holding the following property −

$f (x) = y \Leftrightarrow g (y) = x$

The function f is called invertible, if its inverse function g exists.

Example
- A Function $f : Z \to Z, f (x) = x + 5$ , is invertible since it has the inverse function $g : Z \to Z, g (x) = x - 5$ .
- A Function $f : Z \to Z, f (x) = x^{2}$ is not invertiable since this is not one-to-one as $(- x)^{2} = x^{2}$ .
Composition of Functions

Two functions $f : A \to B$ and $g : B \to C$ can be composed to give a composition $g o f$ . This is a function from A to C defined by $(g o f) (x) = g (f (x))$

Example

Let $f (x) = x + 2$ and $g (x) = 2 x$ , find $(f o g) (x)$ and $(g o f) (x)$ .

Solution

$(f o g) (x) = f (g (x)) = f (2 x + 1) = 2 x + 1 + 2 = 2 x + 3$

$(g o f) (x) = g (f (x)) = g (x + 2) = 2 (x + 2) + 1 = 2 x + 5$

Hence, $(f o g) (x) \neq (g o f) (x)$

Some Facts about Composition
- If f and g are one-to-one then the function $(g o f)$ is also one-to-one.
- If f and g are onto then the function $(g o f)$ is also onto.
- Composition always holds associative property but does not hold commutative property.

rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc.

Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned. The purpose is to analyze these statements either individually or in a composite manner.

Prepositional Logic – Definition

A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.

Some examples of Propositions are given below −

"Man is Mortal", it returns truth value “TRUE”
"12 + 9 = 3 – 2", it returns truth value “FALSE”

The following is not a Proposition −

"A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

Connectives

In propositional logic generally we use five connectives which are −

OR ( $\lor$ )
AND ( $\land$ )
Negation/ NOT ( $\neg$ )
Implication / if-then ( $\to$ )
If and only if ( $\Leftrightarrow$ ).

OR ( $\lor$ ) − The OR operation of two propositions A and B (written as $A \lor B$ ) is true if at least any of the propositional variable A or B is true.