Real Number System
The set of all of real numbers is mentions by R. The real number system is the foundation on which a big part of mathematics with calculus rests. You are well-known with the operations of subtraction, addition, multiplication and division of real numbers and along inequalities. We will recall some properties:
P1 - There R is closed under addition.
P2 - There addition is associative & commutative.
P3 - There R is closed under multiplication.
P4 - There Multiplication is commutative & associative.
P5 -There multiplication is distributive over addition operation.
P6 - For any of two real numbers a & b, either a > b or a < b or a = b.
You are also well known with the following subsets of R.
(a) N, It is the set of natural numbers. (b) Z, it is the set of integers.
(c) Q, it is the set of rational numbers.
Definition 1
If x is a real number, its absolute value, mention by | x | will defined as
| x | = x if x ≥ 0
= - x if x < 0
For instances
| 5 | = 5
| - 5 | = 5
Described theorem (without proof) gives some important properties of | x |.
Theorem 1
If x & y are real numbers, then
(a) | x | = max {- x, x}
(b) | x | = | - x |
(c) | x |2 = x2 = | - x |2
(d) | x + y | ≤ | x | + | y |
(e) | x - y | ≥ | | x | - | y | |
Definition 2
Assume S be a non-empty subset of R. An element u ∈ R is called upon to be upper bound of S if u ≥ x for all of x ∈ S. If S contains an upper bound, we can say S is bounded above. On same lines we can described a lower bound for a non-empty set S to be a real number v like v ≤ x for all of x ∈ S and we state that the set S is bounded below.