real number system proof

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..

We have, (a + b) n = (a + b) (a + b) ....... n times. The terms on the RHS are obtained by taking one letter from each factor and multiplying them together. Choosing 'a' from all the factors, we get the term a n..

A system of linear equations is said to be consistent if it has a solution. This means that the solution satisfies all the equations in the system simultaneously. If a system of linear equations has no solution, then it is said to be inconsiste..

Proof by mathematical induction requires the fulfilment of both the conditions (i) and (ii) as stated above. Condition (i) creates the basis for carrying out induction and condition (ii) gives us the right to make a general assertion or of an automatic extensi..

If x and y are real numbers, then x + iy is called a complex number. x is called the real part and y is called the imaginary part. The complex number x + iy is also written as an ordered pair (x, y) and is denoted by z. i.e., z = x + iy The positive..

For any rational number n, We accept this expansion without proof..

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 i..