Question 1:

a) Make truth table for) → (q ∧ ~ ) ∧ (∼ ∨∼ )) ∼ → (∼ r ∧ q) ∧ (∼ ∨∼ )(i)answer:

b) If A = {1, 2, 3, 4, 5,6,7,8, 9} B {1, 3, 5, 6, 7, 10,12,15}and C {1, 2,3, 10,12,15,45,57}Then (A ∩ B)∆C

c) Write down suitable mathematical statement that can be represented by the followingSymbolic properties.

(i) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ALL OF Z}(ii) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ANY ONE OF Z}

Question 2:

a) ℎ ℎ ? ℎ ℎ ℎ2 ≥ + 1. ) ℎ ℎ ℎ √17 .In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are

assumed to be true, then some mathematical statement is necessarily true. Proofs are obtainedfrom deductive reasoning, rather than from inductive or empirical arguments; a proof mustdemonstrate that a statement is always true (occasionally by listing all possible cases and showingthat it holds in each), rather than enumerate many confirmatory cases. An unproven propositionthat is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits someambiguity. In fact, the vast majority of proofs in written mathematics can be considered asapplications of rigorous informal logic. Purely formal proofs, written in symbolic language instead ofnatural language, are considered in proof theory. The distinction between formal and informalproofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). Thephilosophy of mathematics is concerned with the role of language and logic in proofs, andmathematics as a language.

c) Write down suitable mathematical statement that can be represented by the followingSymbolic properties.

(i) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ALL OF Z}(ii) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ANY ONE OF Z}

Question 2:

a) ℎ ℎ ? ℎ ℎ ℎ2 ≥ + 1. ) ℎ ℎ ℎ √17 .In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are

assumed to be true, then some mathematical statement is necessarily true. Proofs are obtainedfrom deductive reasoning, rather than from inductive or empirical arguments; a proof mustdemonstrate that a statement is always true (occasionally by listing all possible cases and showingthat it holds in each), rather than enumerate many confirmatory cases. An unproven propositionthat is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits someambiguity. In fact, the vast majority of proofs in written mathematics can be considered asapplications of rigorous informal logic. Purely formal proofs, written in symbolic language instead ofnatural language, are considered in proof theory. The distinction between formal and informalproofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). Thephilosophy of mathematics is concerned with the role of language and logic in proofs, andmathematics as a language.

c) Write down suitable mathematical statement that can be represented by the followingSymbolic properties.

(i) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ALL OF Z}(ii) ANSWER :{ ALL OF X} {ANY ONE OF Y} {ANY ONE OF Z}

Question 2:

a) ℎ ℎ ? ℎ ℎ ℎ2 ≥ + 1. ) ℎ ℎ ℎ √17 .In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are

assumed to be true, then some mathematical statement is necessarily true. Proofs are obtainedfrom deductive reasoning, rather than from inductive or empirical arguments; a proof mustdemonstrate that a statement is always true (occasionally by listing all possible cases and showingthat it holds in each), rather than enumerate many confirmatory cases. An unproven propositionthat is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits someambiguity. In fact, the vast majority of proofs in written mathematics can be considered asapplications of rigorous informal logic. Purely formal proofs, written in symbolic language instead ofnatural language, are considered in proof theory. The distinction between formal and informalproofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). Thephilosophy of mathematics is concerned with the role of language and logic in proofs, andmathematics as a language.

) ℎ ℎ ℎ √17 .

c) Explain concept of function with the help of an example? What is relation? Explain following typesof relation with example:

i) Reflexiveii) Symmetriciii) Transitive

When one thing depends on another, as for example the area of a circle depends on the radius -- inthe sense that when the radius changes, the area will change -- then we say that the first is a"function" of the other. The area of a circle is a function of -- it depends on -- the radius.

Mathematically:

A rule that relates two variables, typically x and y, is called a function

If to each value of x the rule assigns one and only one value of y.

When that is the case, we say that y is a function of x.

Thus a "function" must be single valued ("one and only one").

For Example,= 2 + 3To each value of x there is a unique value of y

Examples:

Question 3

a) A Surveys among the players of cricket club, 20 players are pre batsman, 10 players are purebowler, 40 players are all rounder, and 3 players are wicket keeper batsman. Find thefollowings:

i) How many players can either bat or bowl?ii) How many players can bowl?iii) How many players can bat?

b) If p and q are statements, show whether the statement

Question 4:

a) Make logic circuit for the following Boolean expressions:

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematicalstructures into other concepts, theorems or structures, in a one-to-one fashion, often (but notalways) by means of an involution operation: if the dual of A is B, then the dual of B is A. Suchinvolutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues'theorem in projective geometry is self-dual in this sense.In mathematical contexts, duality has numerous meanings although it is "a very pervasive andimportant concept in (modern) mathematics" and "an important general theme that hasmanifestations in almost every area of mathematics."Many mathematical dualities between objects of two types correspond to pairings, bilinear functionsfrom an object of one type and another object of the second type to some family of scalars. Forinstance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spacesto scalars, the duality between distributions and the associated test functions corresponds to thepairing in which one integrates a distribution against a test function, and Poincare dualitycorresponds similarly to intersection number, viewed as a pairing between sub manifolds of a givenmanifold.Duality can also be seen as a function, at least in the realm of vector spaces. There it is allowed toassign to each space its dual space and the pullback construction allows assigning for each arrow f: V→ W, its dual f∗: W∗→ V∗.

Question 5:

Answer: I & II

c) Explain inclusion-exclusion principle with example.

The Inclusion Exclusion Principle

The rule of addition says how many elements are in a union of sets if the sets aremutually disjoint. But there exist cases where one has to determine the number ofelements in a union of sets when some of the sets overlap.

Let us consider the union of two sets A and B. The number of elements in theunion of the two sets varies according to the number of elements in each set thatare common. If A and B have no common elements at all then the number ofelements in their union will be the algebraic sum of the number of elements of Aand the number of elements of B, which is better represented by n(A U B) = n(A) +n(B)

If A and B are exactly the same sets, i.e. if they coincide, then n(AUB) =n(A) .Thereby any formula for the number of elements in a union of two sets mustcontain a reference to the number of elements that are common, n(A B) and alsoreferences to the number of individual elements of the sets (A) and n(B).

To derive a formula for the number of elements in the union of the two sets, thenumber of elements in set A - n(A), which counts the elements in A, the elementsnot in B and the elements that are both in A and B, along with n(B), which countsthe elements in A, the number of elements not in A and the elements that are inboth A and B are considered. By adding n(A) and n(B), the elements inboth A and B are counted twice.

To eliminate this redundancy and get the value for n(AUB), we need to subtract thenumber of elements which are both in A and B, which is represented by theirintersection A B,

Therefore n(AUB)=n(A) + n(B) - n(A B)

Example:Counting the number of elements in a Union.How many integers from 1 to 1000 are either multiples of 3 or multiples of 5

SolutionLet us assume that A = set of all integers from 1-1000 that are multiples of 3Let us assume that B = set of all integers from 1-1000 that are multiples of 5

From this we have A U B = The set of all integers from 1 to 1000 that are multiplesof either 3 or 5 and we also have (A B) = The set of all integers that are bothmultiples of 3 and 5, which also is the set of integers that are multiples of 15.

To use the inclusion/exclusion principle to obtain n(A U B) , weneed n(A),n(B) and n(A B)

From 1 to 1000, every third integer is a multiple of 3,each of this multiple can berepresented as 3p, for any integer p from 1 through 333.

From the above we have that n(A) = 333 for integers 1-1000

Similarly for multiples of 5, each multiple of 5 is of the form 5q for someinteger q from 1 through 200.

From this we have n(B) = 200

For n(A B) , we need to determine the number of multiples of 15 from 1through 1000. Each multiple of 15 is of the form 15r for

some integer r from 1 through 66.Now we have the values for n(A B), which is 66.

From all the above we can determine n(AUB), using the Inclusion/Exclusionprinciple.

n(AUB) = n(A) + n(B) - n(A B)= 333 + 200 - 66

= 467

a) What is pigeonhole principle? Explain its application with the help of an example.

Question 7:

a) Find how many 4 digit numbers are odd?

c) How many different 10 professionals committees can be formed each containing at least 2Project Delivery Managers, at least 2 Technical Architects and 3 Security Experts from list of10 Project Delivery Managers 12 Technical Architects and 5 Security Experts?

Question 7:

a) Find how many 4 digit numbers are odd?

c) How many different 10 professionals committees can be formed each containing at least 2Project Delivery Managers, at least 2 Technical Architects and 3 Security Experts from list of10 Project Delivery Managers 12 Technical Architects and 5 Security Experts?

Question 7:

a) Find how many 4 digit numbers are odd?

c) How many different 10 professionals committees can be formed each containing at least 2Project Delivery Managers, at least 2 Technical Architects and 3 Security Experts from list of10 Project Delivery Managers 12 Technical Architects and 5 Security Experts?

c) Explain concept of permutation with an example. How it is different from combination, explainwith an example?

Question 8:

a) What is Demorgan‟s Law for Boolean algebra? Explain its application with example.

In propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rulesthat are both valid rules of inference. The rules allow the expression of conjunctions anddisjunctions purely in terms of each other via negation.

The rules can be expressed in English as:

The negation of a conjunction is the disjunction of the negations.

The negation of a disjunction is the conjunction of the negations. or informally as:

"not (A and B)" is the same as "(not A) or (not B)"and also,

"not (A or B)" is the same as "(not A) and (not B)".

The rules can be expressed in formal language with two propositions P and Q as:

and

Question 8:

a) What is Demorgan‟s Law for Boolean algebra? Explain its application with example.

In propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rulesthat are both valid rules of inference. The rules allow the expression of conjunctions anddisjunctions purely in terms of each other via negation.

The rules can be expressed in English as:

The negation of a conjunction is the disjunction of the negations.

The negation of a disjunction is the conjunction of the negations. or informally as:

"not (A and B)" is the same as "(not A) or (not B)"and also,

"not (A or B)" is the same as "(not A) and (not B)".

The rules can be expressed in formal language with two propositions P and Q as:

and

Question 8:

a) What is Demorgan‟s Law for Boolean algebra? Explain its application with example.

In propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rulesthat are both valid rules of inference. The rules allow the expression of conjunctions anddisjunctions purely in terms of each other via negation.

The rules can be expressed in English as:

The negation of a conjunction is the disjunction of the negations.

The negation of a disjunction is the conjunction of the negations. or informally as:

"not (A and B)" is the same as "(not A) or (not B)"and also,

"not (A or B)" is the same as "(not A) and (not B)".

The rules can be expressed in formal language with two propositions P and Q as:

and

Where:

¬ is the negation operator (NOT)

is the conjunction operator (AND)

is the disjunction operator (OR) ⇔ is a meta logical symbol meaning "can be replaced in a logical proof with"

Applications of the rules include simplification of logical expressions in computerprograms and digital circuit designs. De Morgan's laws are an example of a moregeneral concept of mathematical duality.

Proof using Set notation and de Morgan’s Law

Where:

¬ is the negation operator (NOT)

is the conjunction operator (AND)

is the disjunction operator (OR) ⇔ is a meta logical symbol meaning "can be replaced in a logical proof with"

Applications of the rules include simplification of logical expressions in computerprograms and digital circuit designs. De Morgan's laws are an example of a moregeneral concept of mathematical duality.

Proof using Set notation and de Morgan’s Law

Where:

¬ is the negation operator (NOT)

is the conjunction operator (AND)

is the disjunction operator (OR) ⇔ is a meta logical symbol meaning "can be replaced in a logical proof with"

Applications of the rules include simplification of logical expressions in computerprograms and digital circuit designs. De Morgan's laws are an example of a moregeneral concept of mathematical duality.

Proof using Set notation and de Morgan’s Law

b) How many “words‟ can be formed using letter of STUDENT using each letter at most once:

i. if each letter must be used,ii. if some or all the letters may be omitted.