This page is particularly meant to define the definition of all basic and necessary concepts taught in SETS. Although the concept in mathematics is enormous so is their definition. Yet we try to explain the very essential definition of nearly all the concepts related to sets used in class 11.
NUMBER SYSTEM
Natural numbers (N):
The natural numbers are the counting numbers used to count the objects or things.
so 1,2,3,......... are called natural numbers or positive integers.
Integers (I or Z):
All the natural numbers along with their negatives as well as number 0 are called integers.
We represent
1. Positive integers = 1, 2, 3, ,........ = natural numbers.
2. Negative integers = .......,-3, -2, -1.
3.Non negative integers = 0, 1, 2, 3,...... = Whole numbers.
4.Non positive integers = .....,-3 ,-2, -1, 0.
5.Prime numbers = 2,5,7........
Prime numbers are those natural numbers which are having exactly two positive factors 1 and number itself.
Composite numbers are those natural numbers which are having more than three positive factors.
1 is an only natural number which is neither prime nor composite.ðŸ’💬
Rational numbers (Q):
A number of the form a/b, where a and b are integers, b ≠ 0 and HCF of a and b is 1, is called a rational number.
note:
Every integer is a rational number as it can be written as q = a/b whenever b = 1.
e.g. 2 = 2/1
All recurring decimal numbers are rational numbers ;
e.g. n = 0.3333... = 1/3.
All terminating numbers in the decimal form are the rational numbers.
e.g. n = 0.5 = 1/2
Irrational numbers( T )
The numbers which are not rational numbers are called an irrational number
e.g. π.
Real numbers ( R ) : 👪
All the rational numbers, as well as irrational numbers, consist of real numbers.
SETS
A set is a well-defined collection of objects.
e.g. set of vowels in the English alphabet, set of all prime numbers, set of all real numbers.
note:
1. objects, elements, and members of a set are synonyms.
2. Sets are usually denoted by capital letters A, B , C, X, T, Z etc..
3.the elements of a set are usually denoted by small letters a,b,c, etc...
Is there any collection that is not a set
yes,
1.The collection of intelligent students in your class,
2.The three most prominent Prime Ministers of India,
3. 10 richest people of the world
All these three collections are not set because these collections are not well defined.
Subsets
A set A is said to be a subset of set B if every element of a set A is also an element of set B.
In notion A ⊂ B ⇔ ∀ a ε A ⇒ a εB.
note:
N ⊂ Z ⊂ Q, Q ⊂ R, T ⊂ R, N ⊄ T
Intervals as subsets of real numbers.
let a, b ε R, and a < b.
1. The set of real numbers { x: a < x < b } is called an open interval and is denoted by (a, b).
This set consists of all real numbers b/w a and b but not a and b themselves.
2. The set of real numbers { x: a ≤ x ≤ b } is called the closed interval and is denoted by [a, b].
this set consists of all real numbers b/w a and b including a and b both.
3.[a, b) = { x: a ≤ x < b } is called open interval from a to b inculding a and excluding b.
4. (a, b] = { x: a < x ≤ b } is called open interval from a to b inculding b andexcluding a.
Power set
The collection of all subsets of the set A is called the power set of A it is denoted by P (A).
e.g if A = { 1 ,2 }. Then P (A) = {Ø,{1},{2}.{1,2}}.
Compliment of a set
Let A be a subset of a universal set U. Then the complement of a set is the set of all those elements of U which don't belong to A.
The complement of a set A = { x: x ε U and x ∉ A } = U - A.
Looking forward to your comments so that I can make even better content for the next definitions.
By: MOHIT KOHLI
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