This document provides information about an assignment for the subject Foundations of Mathematics. It lists the subject code, credit hours, and evaluator ID. It then provides 5 questions related to the course content, such as defining Cauchy's theorem and verifying it for a given function. It also asks students to define logical concepts like tautology, contradiction, negation, and to verify theorems like Lagrange's mean value theorem. The last question asks students to define set theory concepts like the set, null set, subset, power set, and union set. Students are instructed to answer all questions, with 10 mark questions being approximately 400 words each.

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ASSIGNMENT
PROGRAM MCA(REVISED FALL 2012)
SEMESTER FIRST
SUBJECT CODE & NAME MCA1030- FOUNDATION OF MATHEMATICS
CREDIT 4
BK ID B1646
MAX.MARKS 60
Note: Answer all questions. Kindly note that answers for 10 marks questions should be approximately
of 400 words. Each question is followed by evaluation scheme.
1 (i)State Cauchy’s Theorem.
Answer: Cauchy's theorem is a theorem in the mathematics of group theory, named after Augustin
Louis Cauchy. It states that if G is a finite group and p is a prime number dividing theorder of G (the
number of elements in G), then G contains an element of order p. That is, there is x in G so that p is the
lowest non-zero number with xp = e, where e is the identity element.
The theorem is related to Lagrange's theorem, which states that the order of any subgroup of a finite
group G divides the order of G. Cauchy's theorem implies that for any prime divisor p of the order of G,
there is a subgroup of G whose order is p—the
(ii)Verify Cauchy’s Theorem for the following function
푆푖푛푥,표푠푥 푖푛 [0,휋2]

2. Answer: Answer: - A basic concept in the general Cauchy theory is that of winding number or index of a
point with respect to a closed curve not containing the point. In order to make this precise, we need
several preliminary results on logarithm and argument
Q.2 Define Tautology and contradiction. Show that
a) (p q)  (~ p) is a tautology.
b) (p q) (~ p) is a contradiction
Answer: - Tautology: - In logic, a tautology (from the Greek word ταυτολογία) is a formula which is true
in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies
of propositional logic in 1921; (it had been used earlier to refer to rhetorical tautologies, and continues
to be used in that alternate sense). A formula is satisfiable if it is true under at least one interpretation,
and thus a tautology is a formula whose negation is unsati sfiable. Unsatisfiable statements, both
through negation and affirmation, are known
Q.3 State Lagrange’s Theorem. Verify Lagrange’s mean value theorem for the function
f(x) = 3 x2 – 5x + 1 defined in interval [2, 5]
Answer: - Suppose f is a function defined on a closed interval [a,b] (with a<b ) such that the following
two conditions hold:
1. f is a continuous function on the closed interval [a,b](i.e., it is right continuous at a , left continuous at
b , and two-sided continuous at all points in the open interval(a,b) ).
2. f is a differentiable function on the open
Q.4 Define Negation. Write the negation of each of the following conjunctions:
A) Paris is in France and London is in England.
B) 2 + 3 = 5 and 8 < 10.
Answer: - Negation: - In logic, negation, also called logical complement, is an operation that takes a
proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true
when p is false and false when p is true. Negation is thus a unary (single-argument) logical connective. It
may be applied as an operation on propositions, truth values, or semantic values more generally. The
action or logical operation of negating or making negative b : a negative statement, judgment, or
doctrine; especially : a logical proposition formed

3. (b )푥2푦2−푎2(푥2+푦2)=0
Q.5 Find the asymptote parallel to the coordinate axis of the following curves
(i) (푥2+푦2)푥−푎푦2=0
(ii) 푥2푦2−푎2(푥2+푦2)=0
Answer: - (I) (푥2+푦2)푥−푎푦2=0
F(x) = (푥2+푦2)푥−푎푦2

4. (b )푥2푦2−푎2(푥2+푦2)=0
Q.6 Define (I) Set (ii) Null Set (iii) Subset (iv) Power set (v)Union set
Answer: - Set: - In mathematics, a set is a collection of distinct objects, considered as an object in its
own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but
when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one
of the most fundamental concepts in mathematics. In everyday life, we have to deal with the
collections of objects of one kind or the other.
 The collection of even natural numbers less than 12 i.e., of the numbers 2,4,6,8, and 10.
 The collection of vowels in the English alphabet, i.e., of the letters a ,e ,i ,o , u.
 The collection of all students of class