Some subsets of R #some #subset #R
Some important subsets of R.
We define some important subsets of R the set of real numbers, namely
1. The set of natural numbers
2. The set of Whole numbers
3.The set of integers
4.The set of rational numbers
5.The set of irrational numbers in different manner.
Inductive set:
If A is a subset of R is such that
i) 1 Ꜫ A ii) p Ꜫ A implies p+1 Ꜫ A , then A is called an Inductive set.
Examples and counter examples:
1. The set R, of real numbers is an inductive set.
2. the set Z+ of all + ve integers is an inductive set.
3. the set Z- of all -ve integers is not inductive since 1 ∉ Z-
Natural numbers:
The set of all natural numbers is the smallest inductive set or intersection of all inductive sets.
The set of all natural numbers is denoted by N or Z+
Principle of induction (or) principle of mathematical induction
If a statement p(n) of natural numbers is such that
(i) p(1) is true
(ii) p(k) is true p(k+1) is true, then p(n) is true for every n Ꜫ N.
PEANO's AXIOMS:
The axioms for the set N of natural numbers, known as Peano's axioms, are as follows.
1. 1Ꜫ N
2. n ꜪN implies n+1 ꜪN
3. n ꜪN implies n+1 ≠ 1.
4. m, n ꜪN , m+1=n+1 implies m=n.
Well ordering principle:-
Every non-empty subset of N has a least element.
The set of whole numbers :-
The set of natural numbers with the number zero '0' is called the set of whole numbers
and it is denoted by W.
Therefore W={ 0,1,2,...}
Properties of W:
The set W of whole numbers has properties as the set N of natural numbers together with
for aꜪ W , a+0 = a , which is called identity property under addition.
The set of integers:-
The set is called the set of integers. The set of integers is denoted by Z or I.
we have Z = { ...,-3,-2,-1,0,1,2,3,...}
some properties of Z:-
1.The system (Z,+, .) satisfies the field axioms except multiplicative inverse condition i.e. for a ꜪZ, 1/a ∉ Z
2. For p,q Ꜫ Z, p+q, p-q, pq Ꜫ Z.
3. For p, q Ꜫ Z , p/q ∉ Z.
4. The square root of even integer is even if exist
5. The square root of odd integer is odd if exist
The set of rational numbers:
The set { m/n such that m, n ꜪZ & n ≠0} is called the set of rational numbers. It is denoted by Q.
From the above definition , if we see which is 1/3 in the form m/n where n=3 ≠0, according to definition 1/3 is rational but 1/3 is not rational
then how can we define a number as rational?
" A number which has a finite decimal part i.e., the decimal place is terminated after a finite position".
Here the decimal value is stopped with 5, there is no further value after 5.
If we choose =0.3333......., i.e., the decimal value has 3 repeatedly infinite times which is not terminated.
Some properties of Q:
1. For p,q Ꜫ Q, p+q, p-q, pq Ꜫ Q
2. The representation of a rational number is unique.
3. Between two distinct rational numbers a, b there is a rational number.
4. The set of all rational numbers satisfies all field axioms.
The set of irrational numbers:
The set which is not contains the rational numbers is called the set of irrational numbers.
i.e., a number which is not rational is an irrational number.
Ex: Since 1/3 is not rational, hence 1/3 is irrational.
Properties of irrational numbers:
The set of irrationals is denoted by R-Q.
1. If n Ꜫ N is not a perfect square then √n is irrational.
2. The sum and product of two irrational numbers need not always be irrational. Hence the set R-Q not satisfying field axioms.
3. Between two distinct rational numbers there exists an irrational number.
People also see
* basic definitions of real numbers
* introduction to real analysis
#some #subset #R
#natural #number #whole #integer #rational #irrational #real #R-Q
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