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Introduction
The word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a pr..
The word 'Induction' means method of reasoning from individual cases to general ones or from observed instances to unobserved ones. Many important mathematical formulae are such that a result is formed by some means which does not provide for a direct proof. Mathematical Induction is a pr..Proof:
When x = 1, When x = -1 \ Hence (i) is proved. \ Hence (ii) is proved. \ Hence (iii) is prov..
When x = 1, When x = -1 \ Hence (i) is proved. \ Hence (ii) is proved. \ Hence (iii) is prov..Proof:
We have, Replacing q by ix, ..
We have, Replacing q by ix, ..Proof:
Similarly,..
Similarly,..Proof:
When p(x) is divided by x-a, R = p(a) (by remainder theorem) p(x) = (x-a).q(x)+p(a) (Dividend = Divisor x quotient + Remainder Division Algorithm) But p(a) = 0 is given. Hence p(x) = (x-a).q(x) Conversely if x-a is a factor of p(x) then p(a)=0. p(x) = (x-a).q(x) + R If (x-a) is a factor then the..
When p(x) is divided by x-a, R = p(a) (by remainder theorem) p(x) = (x-a).q(x)+p(a) (Dividend = Divisor x quotient + Remainder Division Algorithm) But p(a) = 0 is given. Hence p(x) = (x-a).q(x) Conversely if x-a is a factor of p(x) then p(a)=0. p(x) = (x-a).q(x) + R If (x-a) is a factor then the..Proof:
..
..Proof:
We must prove two statements. If x A and x B then, by the statement "two sets A and B are different if there exists an element which belongs to one set but not to the other" and by hypothesis, A = B is contradicted. Thus A B. Similarly B A. If A B, then there is an eleme..
We must prove two statements. If x A and x B then, by the statement "two sets A and B are different if there exists an element which belongs to one set but not to the other" and by hypothesis, A = B is contradicted. Thus A B. Similarly B A. If A B, then there is an eleme..Proof:
r = (0, 1, 2, 3, . . . n) Hence the total number of subsets i..
r = (0, 1, 2, 3, . . . n) Hence the total number of subsets i..Proof:
The null set f is a subset of every set, in particular f A. By hypothesis, A f . The two conditions imply A = f..
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