Assignment of System Fundamental of
Mathematics
Answer:-
Leibnitz’s Theorem
Statement: Let u(x) and v(x) be nth order differentiable functions of x
and
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Question 2:- Define Tautology and contradiction. Show that
a) for (p Γ q) Γ (~ p) is a
tautology.
b) is (p
Γ q) Γ (~ p) a contradiction
Answer:-
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Definition
of Tautologies and Contradiction:-
A statement is said to be a tautology if it is true for all
logical possibilities. In
other words, a statement is called tautology if its truth
value is T and only T
in the last column of its truth table. Analogously, a
statement is said to be a
contradiction if it is false for all logical
possibilities. In other words, a
statement is called contradiction if its truth value is F
and only F in the last
column of its truth table. A straight forward method to
determine whether a
given statement is tautology (or contradiction)
Solution of Tautology:-
The truth table for (p Γ q) Γ (~ p) is given by
Since the truth table for (p Γ q) Γ (~ p) contains only T in the last column,
it follows that (p Γ q) Γ (~ p) is a tautology.
Solution of contradiction:-
Truth table for (p Γ q) Γ (~ p) and observe
that it contains only F in the last column. Therefore, (p
Γ q) Γ (~ p) is a
contradiction.
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Question 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]
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Answer:-
Lagrange’s Theorem
Solution: We have , f(x) = 3 x2 – 5x + 1 where , x Ξ΅ [2, 5] .
(1) f(x) is a polynomial function , hence continuous in the
interval [2, 5] .
(2) f(x) is a polynomial function , hence differentiable in
the interval (2, 5) .
(3) f(5) = 3 (5) 2 –
5 × 5 + 1 = 51 , f(2) = 3 (2) 2 – 5 ×
2 + 1 = 3 .
Also , f (x) = 6 x – 5 => f’(c) = 6 c – 5 .
Now ,f (c) = [f(b) – f(a)]/(b – a)
Or , 6 c – 5 = [f(5) – f(2)]/(5 – 2) = (51 – 3)(5 – 2) =
48/3 = 16
Or , 6 c = 16 + 5 = 21 => c = 21/6 Ξ΅ (2, 5)
Hence , Lagrange’s mean value theorem is verified.
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Question 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.
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Answer:-
Definition of Negation:-
An assertion
that a statement fails or denial of a statement is called the
negation of the
statement. The negation of a statement is generally formed
by introducing
the word “not” at some proper place in the statement or by
prefixing the
statement with “It is not the case that” or “It is false that”.
The negation of
a statement p in symbolic form is written as “~ p”.
Example:-
Write the negation of the
statement
p :New Delhi is a city.
Solution:- The
negation of p is given by
~ p :New Delhi is not a city
or ~ p : It
is not the case that New Delhi is a city.
or ~ p : It
is false that New Delhi is a city
Solution:
(a) Write p :Paris is in France and q :London is
in England.
Then, the conjunction in (a) is given by p Γ q.
Now ~ p :Paris is not in France, and
~ q :London is not in England.
Therefore, using (D7), negation of p Γq is given by
~( p Γ q) = Paris is
not in France or London is not in England.
(b) Write p : 2+3 = 5 and q :8 < 10.
Then the conjunction in (b) is
given by p Γ q.
Now ~ p : 2 + 3 ≠ 5 and ~q:8</10
Then, using (D7), negation
of p Γ q is given by
~ (p Γ q) = 2 + 3 ≠ 5 or (8</10 ).
Question 5:- Find the
asymptote parallel to the coordinate axis of the following curves
(i) (π₯2+π¦2)−ππ¦2=0
(ii) π₯2π¦2−π2(π₯2+π¦2)=0
Answer:-
Solution:-
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Questions 6:- Define
(i) Set (ii)Null Set(iii)
Subset(iv)Power set (v)Union set
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Answer:-
i.Sets
In every day life, we often speak of
collection of objects of a particular kind
such as pack of
cards, a herd of cattle, a crowd of people, cricket team, etc.
In mathematics
also, we come across various collections, for example,
collection of
natural numbers, points in plane, prime numbers, etc.
There are two methods of representing a set:
i) In roster form:- all the elements of a set are listed,
the elements being
separated by
commas and are enclosed within braces { }. For example,
the set of all
even positive integers less than 7 is described in roster
form as {2, 4,
6}.
ii) In set builder form:- all the elements of a set possess a
single common
property which
is not possessed by any element outside the set. For
example, in the
set “{a, e, i, o, u}” all the elements possess a common
property, each
of them is a vowel in the English alphabet and no other
letter
possesses this property. Denoting this set by V, we write
V = {x : x is a
vowel in the English alphabet}.
ii.The Empty Set
Definition:-
A set which does not contain any element is
called an empty set
or null set or
the void set.
According to
this definition B is an empty set while A is not. The empty set is
denoted by the
symbol ‘Γ’.
Example:-
I ) Let P = {x:
1 < x < 2, x is a natural number }.
Then P is an empty set, because there is no
natural number between
1 and 2.
iii.Subsets
Definition: If every element of a set A is also an
element of a set B, then A
is called a
subset of B or A is contained in B. We write it as A C B.
If at least one
element of A does not belong to B, then A is not a subset of
B.
Examples:-
i The set Q of
rational numbers is a subset of the set R of real numbers
and we write Q
C R.
iv.Power Set:-
Definition: The collection of all subsets of a set
A is called the power set of
A. It is
denoted by P(A). In P(A), every element is a set.
Example:- we
found all the subsets of the set {1, 2}, viz.,
, {1}, {2} and
{1, 2}. The set of all these four subsets is called the power set
of {1, 2}.
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v. Union of Sets
Definition:- The union of two sets A and B is the
set C which consists of all
those elements
which are either in A or in B (including those which are in
both).
Let A and B be any two sets. The union of A
and B is
the set which
consists of all the elements of A as well as the elements of B,
the common
elements being taken only once. The symbol ‘αΉΈ’ is used to
denote the
union.
Example:-
‘A union B’
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