MATHEMATICS1A STUDY MATERIAL
1st Year Maths IA Study Material 
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, A={a,e,i,o,u}
Example 2 − Set of odd numbers less than 10, B={1,3,5,7,9}
Set Builder Notation
The set is defined by specifying a property that elements of the set have in common. The set is described as A={x:p(x)}
Example 1 − The set {a,e,i,o,u} is written as −
A={x:x is a vowel in English alphabet}
Example 2 − The set {1,3,5,7,9} is written as −
B={x:1≤x<10 and (x%2)≠0}
If an element x is a member of any set S, it is denoted by x∈S and if an element y is not a member of set S, it is denoted by y∉S.
Example − If S={1,1.2,1.7,2},1∈S but 1.5∉S
Some Important Sets
N − the set of all natural numbers = {1,2,3,4,.....}
Z − the set of all integers = {.....,−3,−2,−1,0,1,2,3,.....}
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 S, 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 − {1,4,3,5}=4,{1,2,3,4,5,…}=∞
If there are two sets X and Y,
 X=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.
 X≤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.
 X<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.
 If X≤Y and X≤Y then X=Y. 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 − S={xx∈N and 70>x>50}
Infinite Set
A set which contains infinite number of elements is called an infinite set.
Example − S={xx∈N and x>10}
Subset
A set X is a subset of set Y (Written as X⊆Y) if every element of X is an element of set Y.
Example 1 − Let, X={1,2,3,4,5,6} and Y={1,2}. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y⊆X.
Example 2 − Let, X={1,2,3} and Y={1,2,3}. 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 Y⊆X.
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 X⊂Y) if every element of X is an element of set Y and X<Y.
Example − Let, X={1,2,3,4,5,6} and Y={1,2}. Here set Y⊂X since all elements in Y are contained in X too and X has at least one element is more than set Y.
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 U.
Example − We may define U as the set of all animals on earth. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, 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 − S={xx∈N and 7<x<8}=∅
Singleton Set or Unit Set
Singleton set or unit set contains only one element. A singleton set is denoted by {s}.
Example − S={xx∈N, 7<x<9} = {8}
Equal Set
If two sets contain the same elements they are said to be equal.
Example − If A={1,2,6} and B={6,1,2}, 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 A={1,2,6} and B={16,17,22}, they are equivalent as cardinality of A is equal to the cardinality of B. i.e. A=B=3
Overlapping Set
Two sets that have at least one common element are called overlapping sets.
In case of overlapping sets −
 n(A∪B)=n(A)+n(B)−n(A∩B)
 n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
 n(A)=n(A−B)+n(A∩B)
 n(B)=n(B−A)+n(A∩B)
Example − Let, A={1,2,6} and B={6,12,42}. 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 −
 n(A∩B)=∅
 n(A∪B)=n(A)+n(B)
Example − Let, A={1,2,6} and B={7,9,14}, 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
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 A∪B) is the set of elements which are in A, in B, or in both A and B. Hence, A∪B={xx∈A OR x∈B}.
Example − If A={10,11,12,13} and B = {13,14,15}, then A∪B={10,11,12,13,14,15}. (The common element occurs only once)
Set Intersection
The intersection of sets A and B (denoted by A∩B) is the set of elements which are in both A and B. Hence, A∩B={xx∈A AND x∈B}.
Example − If A={11,12,13} and B={13,14,15}, then A∩B={13}.
Set Difference/ Relative Complement
The set difference of sets A and B (denoted by A–B) is the set of elements which are only in A but not in B. Hence, A−B={xx∈A AND x∉B}.
Example − If A={10,11,12,13} and B={13,14,15}, then (A−B)={10,11,12} and (B−A)={14,15}. Here, we can see (A−B)≠(B−A)
Complement of a Set
The complement of a set A (denoted by A′) is the set of elements which are not in set A. Hence, A′={xx∉A}.
More specifically, A′=(U−A) where U is a universal set which contains all objects.
Example − If A={xx belongstosetofoddintegers} then A′={yy doesnotbelongtosetofoddintegers}
Cartesian Product / Cross Product
The Cartesian product of n number of sets A1,A2,…An denoted as A1×A2⋯×An can be defined as all possible ordered pairs (x1,x2,…xn) where x1∈A1,x2∈A2,…xn∈An
Example − If we take two sets A={a,b} and B={1,2},
The Cartesian product of A and B is written as − A×B={(a,1),(a,2),(b,1),(b,2)}
The Cartesian product of B and A is written as − B×A={(1,a),(1,b),(2,a),(2,b)}
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 2n. Power set is denoted as P(S).
Example −
For a set S={a,b,c,d} let us calculate the subsets −
 Subsets with 0 elements − {∅} (the empty set)
 Subsets with 1 element − {a},{b},{c},{d}
 Subsets with 2 elements − {a,b},{a,c},{a,d},{b,c},{b,d},{c,d}
 Subsets with 3 elements − {a,b,c},{a,b,d},{a,c,d},{b,c,d}
 Subsets with 4 elements − {a,b,c,d}
Hence, P(S)=
{{∅},{a},{b},{c},{d},{a,b},{a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d}}
P(S)=24=16
Note − The power set of an empty set is also an empty set.
P({∅})=20=1
Partitioning of a Set
Partition of a set, say S, is a collection of n disjoint subsets, say P1,P2,…Pn that satisfies the following three conditions −
 Pi does not contain the empty set.[Pi≠{∅} for all 0<i≤n]
 The union of the subsets must equal the entire original set.[P1∪P2∪⋯∪Pn=S]
 The intersection of any two distinct sets is empty.[Pa∩Pb={∅}, for a≠b where n≥a,b≥0]
Example
Let S={a,b,c,d,e,f,g,h}
One probable partitioning is {a},{b,c,d},{e,f,g,h}
Another probable partitioning is {a,b},{c,d},{e,f,g,h}
Bell Numbers
Bell numbers give the count of the number of ways to partition a set. They are denoted by Bn where n is the cardinality of the set.
Example −
Let S={1,2,3}, n=S=3
The alternate partitions are −
1. ∅,{1,2,3}
2. {1},{2,3}
3. {1,2},{3}
4. {1,3},{2}
5. {1},{2},{3}
Hence B3=5
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 xRyxRy or R(x,y)R(x,y)) is a subset of the Cartesian product x×yx×y. If the ordered pair of G is reversed, the relation also changes.
Generally an nary relation R between sets A1,…, and AnA1,…, and An is a subset of the nary product A1×⋯×AnA1×⋯×An. The minimum cardinality of a relation R is Zero and maximum is n2n2 in this case.
A binary relation R on a single set A is a subset of A×AA×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)∈RforsomeyinB}{x(x,y)∈RforsomeyinB}
 The range of R, Ran(R), is the set {y(x,y)∈RforsomexinA}{y(x,y)∈RforsomexinA}
Examples
Let, A={1,2,9}A={1,2,9} and B={1,3,7}B={1,3,7}
 Case 1 − If relation R is 'equal to' then R={(1,1),(3,3)}R={(1,1),(3,3)}Dom(R) = {1,3},Ran(R)={1,3}{1,3},Ran(R)={1,3}
 Case 2 − If relation R is 'less than' then R={(1,3),(1,7),(2,3),(2,7)}R={(1,3),(1,7),(2,3),(2,7)}Dom(R) = {1,2},Ran(R)={3,7}{1,2},Ran(R)={3,7}
 Case 3 − If relation R is 'greater than' then R={(2,1),(9,1),(9,3),(9,7)}R={(2,1),(9,1),(9,3),(9,7)}Dom(R) = {2,9},Ran(R)={1,3,7}{2,9},Ran(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)}R={(1,1),(1,2),(3,2)} on set S={1,2,3}S={1,2,3}, it can be represented by the following graph −
Types of Relations
 The Empty Relation between sets X and Y, or on E, is the empty set ∅∅
 The Full Relation between sets X and Y is the set X×YX×Y
 The Identity Relation on set X is the set {(x,x)x∈X}{(x,x)x∈X}
 The Inverse Relation R' of a relation R is defined as − R′={(b,a)(a,b)∈R}R′={(b,a)(a,b)∈R}Example − If R={(1,2),(2,3)}R={(1,2),(2,3)} then R′R′ will be {(2,1),(3,2)}{(2,1),(3,2)}
 A relation R on set A is called Reflexive if ∀a∈A∀a∈A is related to a (aRa holds)Example − The relation R={(a,a),(b,b)}R={(a,a),(b,b)} on set X={a,b}X={a,b}is reflexive.
 A relation R on set A is called Irreflexive if no a∈Aa∈A is related to a (aRa does not hold).Example − The relation R={(a,b),(b,a)}R={(a,b),(b,a)} on set X={a,b}X={a,b}is irreflexive.
 A relation R on set A is called Symmetric if xRyxRy implies yRxyRx, ∀x∈A∀x∈A and ∀y∈A∀y∈A.Example − The relation R={(1,2),(2,1),(3,2),(2,3)}R={(1,2),(2,1),(3,2),(2,3)} on set A={1,2,3}A={1,2,3} is symmetric.
 A relation R on set A is called AntiSymmetric if xRyxRy and yRxyRximplies x=y∀x∈Ax=y∀x∈A and ∀y∈A∀y∈A.Example − The relation R={(x,y)→Nx≤y}R={(x,y)→Nx≤y} is antisymmetric since x≤yx≤y and y≤xy≤x implies x=yx=y.
 A relation R on set A is called Transitive if xRyxRy and yRzyRz implies xRz,∀x,y,z∈AxRz,∀x,y,z∈A.Example − The relation R={(1,2),(2,3),(1,3)}R={(1,2),(2,3),(1,3)} on set A={1,2,3}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)}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}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 onetoone relationship between its xvalues and yvalues.
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 yvalues 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 xvalue for every one yvalue.
Graphs
We can also use graphs to identify functions by plotting ordered pairs onto a Cartesian Coordinate System, where the xvalues are on the horizontal line and the yvalues 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 ffunction. 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 xvalue or domain is called the independent variable. Here we'll use 2, 1, 0, 1, and 2 for the domain.
The range or yvalue is called the dependent variable because it depends on what we use for the xterm.
A function or mapping (Defined as f:X→Yf:X→Y) is a relationship from elements of one set X to elements of another set Y (X and Y are nonempty 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∈Xx∈X, there exists a unique y∈Yy∈Y such that (x,y)∈R(x,y)∈R. ‘x’ is called preimage 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 / Onetoone function
A function f:A→Bf:A→B is injective or onetoone function if for every b∈Bb∈B, there exists at most one a∈Aa∈A such that f(s)=tf(s)=t.
This means a function f is injective if a1≠a2a1≠a2 implies f(a1)≠f(a2)f(a1)≠f(a2).
Example
 f:N→N,f(x)=5xf:N→N,f(x)=5x is injective.
 f:N→N,f(x)=x2f:N→N,f(x)=x2 is injective.
 f:R→R,f(x)=x2f:R→R,f(x)=x2 is not injective as (−x)2=x2(−x)2=x2
Surjective / Onto function
A function f:A→Bf:A→B is surjective (onto) if the image of f equals its range. Equivalently, for every b∈Bb∈B, there exists some a∈Aa∈A such that f(a)=bf(a)=b. This means that for any y in B, there exists some x in A such that y=f(x)y=f(x).
Example
 f:N→N,f(x)=x+2f:N→N,f(x)=x+2 is surjective.
 f:R→R,f(x)=x2f:R→R,f(x)=x2 is not surjective since we cannot find a real number whose square is negative.
Bijective / Onetoone Correspondent
A function f:A→Bf:A→B is bijective or onetoone correspondent if and only if f is both injective and surjective.
Problem
Prove that a function f:R→Rf:R→R defined by f(x)=2x–3f(x)=2x–3 is a bijective function.
Explanation − We have to prove this function is both injective and surjective.
If f(x1)=f(x2)f(x1)=f(x2), then 2x1–3=2x2–32x1–3=2x2–3 and it implies that x1=x2x1=x2.
Hence, f is injective.
Here, 2x–3=y2x–3=y
So, x=(y+5)/3x=(y+5)/3 which belongs to R and f(x)=yf(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 onetoone corresponding function f:A→Bf:A→B, is the function g:B→Ag:B→A, holding the following property −
f(x)=y⇔g(y)=xf(x)=y⇔g(y)=x
The function f is called invertible, if its inverse function g exists.
Example
 A Function f:Z→Z,f(x)=x+5f:Z→Z,f(x)=x+5, is invertible since it has the inverse function g:Z→Z,g(x)=x−5g:Z→Z,g(x)=x−5.
 A Function f:Z→Z,f(x)=x2f:Z→Z,f(x)=x2 is not invertiable since this is not onetoone as (−x)2=x2(−x)2=x2.
Composition of Functions
Two functions f:A→Bf:A→B and g:B→Cg:B→C can be composed to give a composition gofgof. This is a function from A to C defined by (gof)(x)=g(f(x))(gof)(x)=g(f(x))
Example
Let f(x)=x+2f(x)=x+2 and g(x)=2xg(x)=2x, find (fog)(x)(fog)(x) and (gof)(x)(gof)(x).
Solution
(fog)(x)=f(g(x))=f(2x+1)=2x+1+2=2x+3(fog)(x)=f(g(x))=f(2x+1)=2x+1+2=2x+3
(gof)(x)=g(f(x))=g(x+2)=2(x+2)+1=2x+5(gof)(x)=g(f(x))=g(x+2)=2(x+2)+1=2x+5
Hence, (fog)(x)≠(gof)(x)(fog)(x)≠(gof)(x)
Some Facts about Composition
 If f and g are onetoone then the function (gof)(gof) is also onetoone.
 If f and g are onto then the function (gof)(gof) 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 (∨∨)
 AND (∧∧)
 Negation/ NOT (¬¬)
 Implication / ifthen (→→)
 If and only if (⇔⇔).
OR (∨∨) − The OR operation of two propositions A and B (written as A∨BA∨B) is true if at least any of the propositional variable A or B is true.
The truth table is as follows −
A  B  A ∨ B 

True  True  True 
True  False  True 
False  True  True 
False  False  False 
AND (∧∧) − The AND operation of two propositions A and B (written as A∧BA∧B) is true if both the propositional variable A and B is true.
The truth table is as follows −
A  B  A ∧ B 

True  True  True 
True  False  False 
False  True  False 
False  False  False 
Negation (¬¬) − The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false.
The truth table is as follows −
A  ¬ A 

True  False 
False  True 
Implication / ifthen (→→) − An implication A→BA→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true.
The truth table is as follows −
A  B  A → B 

True  True  True 
True  False  False 
False  True  True 
False  False  True 
If and only if (⇔⇔) − A⇔BA⇔B is biconditional logical connective which is true when p and q are same, i.e. both are false or both are true.
The truth table is as follows −
A  B  A ⇔ B 

True  True  True 
True  False  False 
False  True  False 
False  False  True 
Tautologies
A Tautology is a formula which is always true for every value of its propositional variables.
Example − Prove [(A→B)∧A]→B[(A→B)∧A]→B is a tautology
The truth table is as follows −
A  B  A → B  (A → B) ∧ A  [( A → B ) ∧ A] → B 

True  True  True  True  True 
True  False  False  False  True 
False  True  True  False  True 
False  False  True  False  True 
As we can see every value of [(A→B)∧A]→B[(A→B)∧A]→B is "True", it is a tautology.
Contradictions
A Contradiction is a formula which is always false for every value of its propositional variables.
Example − Prove (A∨B)∧[(¬A)∧(¬B)](A∨B)∧[(¬A)∧(¬B)] is a contradiction
The truth table is as follows −
A  B  A ∨ B  ¬ A  ¬ B  (¬ A) ∧ ( ¬ B)  (A ∨ B) ∧ [( ¬ A) ∧ (¬ B)] 

True  True  True  False  False  False  False 
True  False  True  False  True  False  False 
False  True  True  True  False  False  False 
False  False  False  True  True  True  False 
As we can see every value of (A∨B)∧[(¬A)∧(¬B)](A∨B)∧[(¬A)∧(¬B)] is “False”, it is a contradiction.
Contingency
A Contingency is a formula which has both some true and some false values for every value of its propositional variables.
Example − Prove (A∨B)∧(¬A)(A∨B)∧(¬A) a contingency
The truth table is as follows −
A  B  A ∨ B  ¬ A  (A ∨ B) ∧ (¬ A) 

True  True  True  False  False 
True  False  True  False  False 
False  True  True  True  True 
False  False  False  True  False 
As we can see every value of (A∨B)∧(¬A)(A∨B)∧(¬A) has both “True” and “False”, it is a contingency.
Propositional Equivalences
Two statements X and Y are logically equivalent if any of the following two conditions hold −
 The truth tables of each statement have the same truth values.
 The biconditional statement X⇔YX⇔Y is a tautology.
Example − Prove ¬(A∨B)and[(¬A)∧(¬B)]¬(A∨B)and[(¬A)∧(¬B)] are equivalent
Testing by 1^{st} method (Matching truth table)
A  B  A ∨ B  ¬ (A ∨ B)  ¬ A  ¬ B  [(¬ A) ∧ (¬ B)] 

True  True  True  False  False  False  False 
True  False  True  False  False  True  False 
False  True  True  False  True  False  False 
False  False  False  True  True  True  True 
Here, we can see the truth values of ¬(A∨B)and[(¬A)∧(¬B)]¬(A∨B)and[(¬A)∧(¬B)] are same, hence the statements are equivalent.
Testing by 2^{nd} method (Biconditionality)
A  B  ¬ (A ∨ B )  [(¬ A) ∧ (¬ B)]  [¬ (A ∨ B)] ⇔ [(¬ A ) ∧ (¬ B)] 

True  True  False  False  True 
True  False  False  False  True 
False  True  False  False  True 
False  False  True  True  True 
As [¬(A∨B)]⇔[(¬A)∧(¬B)][¬(A∨B)]⇔[(¬A)∧(¬B)] is a tautology, the statements are equivalent.
Inverse, Converse, and Contrapositive
Implication / ifthen (→)(→) is also called a conditional statement. It has two parts −
 Hypothesis, p
 Conclusion, q
As mentioned earlier, it is denoted as p→qp→q.
Example of Conditional Statement − “If you do your homework, you will not be punished.” Here, "you do your homework" is the hypothesis, p, and "you will not be punished" is the conclusion, q.
Inverse − An inverse of the conditional statement is the negation of both the hypothesis and the conclusion. If the statement is “If p, then q”, the inverse will be “If not p, then not q”. Thus the inverse of p→qp→q is ¬p→¬q¬p→¬q.
Example − The inverse of “If you do your homework, you will not be punished” is “If you do not do your homework, you will be punished.”
Converse − The converse of the conditional statement is computed by interchanging the hypothesis and the conclusion. If the statement is “If p, then q”, the converse will be “If q, then p”. The converse of p→qp→q is q→pq→p.
Example − The converse of "If you do your homework, you will not be punished" is "If you will not be punished, you do not do your homework”.
Contrapositive − The contrapositive of the conditional is computed by interchanging the hypothesis and the conclusion of the inverse statement. If the statement is “If p, then q”, the contrapositive will be “If not q, then not p”. The contrapositive of p→qp→q is ¬q→¬p¬q→¬p.
Example − The Contrapositive of " If you do your homework, you will not be punished” is "If you are not punished, then you do not do your homework”.
Duality Principle
Duality principle states that for any true statement, the dual statement obtained by interchanging unions into intersections (and vice versa) and interchanging Universal set into Null set (and vice versa) is also true. If dual of any statement is the statement itself, it is said selfdual statement.
Example − The dual of (A∩B)∪C(A∩B)∪C is (A∪B)∩C(A∪B)∩C
Normal Forms
We can convert any proposition in two normal forms −
 Conjunctive normal form
 Disjunctive normal form
Conjunctive Normal Form
A compound statement is in conjunctive normal form if it is obtained by operating AND among variables (negation of variables included) connected with ORs. In terms of set operations, it is a compound statement obtained by Intersection among variables connected with Unions.
Examples
 (A∨B)∧(A∨C)∧(B∨C∨D)(A∨B)∧(A∨C)∧(B∨C∨D)
 (P∪Q)∩(Q∪R)(P∪Q)∩(Q∪R)
Disjunctive Normal Form
A compound statement is in conjunctive normal form if it is obtained by operating OR among variables (negation of variables included) connected with ANDs. In terms of set operations, it is a compound statement obtained by Union among variables connected with Intersections.
Examples
 (A∧B)∨(A∧C)∨(B∧C∧D)(A∧B)∨(A∧C)∨(B∧C∧D)
 (P∩Q)∪(Q∩R)
Mathematics 1A important questions
Maths_1A__Chapter_wise_important_Questions
1a
mathematical induction