definition of real numbers proof

From the definition of inverse of a matrix, we have (AB)(AB) - 1 = I or A - 1 (AB)(AB) - 1 = A - 1 I (Pre-multiplying both sides by A - 1 ) or (A - 1 A) B (AB) - 1 = A - 1 (Since A - 1 I = A - 1 ) or I B (AB) - 1 = A - 1 or B (AB) - 1 = A - 1 or (B - 1 B)(AB) - ..

When x = 1, When x = -1 \ Hence (i) is proved. \ Hence (ii) is proved. \ Hence (iii) is prov..

We have, Replacing q by ix, ..

Similarly,..

Let p(x) be a polynomial divided by (x-a). Let q(x) be the quotient and R be the remainder. By division algorithm, Dividend = (Divisor x quotient) + Remainder p(x) = q(x) . (x-a) + R Substitute x = a, p(a) = q(a) (a-a) + R p(a) = R (a - a = 0, 0 - q (a) = 0) Hence Remainder = p(..

Let A be any set. In order to prove that f A we must show that there is no element of f which is not present in A. And since f contains no element at all, no such element can be found out. Hence f A...

Real Numbers - The union of the set of rational numbers and irrational numbers forms the set of real numbers. Q = {rational numbers} = {irrational numbers} Then = R = {real numbers..

Every set A is a subset of universal set U since, by definition all elements of A belong to U. Also the null set f A..