Today is our topic of discussion Set of Higher Mathematics .
Set of Higher Mathematics
Any well-defined collection of objects of the real world or of the conceptual realm is called Set. For example, S = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100} denotes the set of square of natural numbers which are less than 10.
The process of expressing set in this method is called Tabular Method. Each of the objects forming a set is called Element of that set. If x is an element of a set A then we write x Є A and if x is not an element of a set A then we write x A. The aforementioned set S is written as S = {x: x is square number not greater than 100 }. This method is called Set Builder Method.
Universal Set
Suppose,
S = {x:x is a positive integer and 5x ≤ 16}
T = {x: x is a positive integer and x² < 20}
P = {x: x is a positive integer and √x ≤ 2}
The elements of these three sets are formed by the set U = {x: x is a positive integer). U can be considered as the universal set for sets S, T, P.
If all the sets under discussion are included in a particular set, that particular set is called the universal set.
Some Special Number Sets
N = {1, 2, 3,…}, that is the set of all natural numbers or positive integers.
Z = {…, -2, -1, 0, 1, 2, 3, … }, that is the set of all integers.
Q = {x: x = Where p is any integer and q is any positive integers} that is the set of all rational numbers.
R = {x: x is a real number} So the set of all real numbers.
Subset
If A and B are sets then A will be called the subset of B if and only if every element of A are elements of B and it will be expressed as AC B. For example, A = {2, 3} is the subset of B = {2, 3, 5, 7}. If A is not the subset of B then A Z B is written. For example, A = {1,3} is not the subset of B = {2, 3, 5, 7}.
Empty Set
Sometimes such a set is considered which has no element. This kind of set is regarded as empty set and expressed as Ø or {}.
Equality of Sets
If A and B sets have the same elements then A and B are regarded to be equal and it is expressed as A = B For example, A = {1, 2, 3, 4} , B = {1, 2, 2, 3, 4, 4, 4} . It is to be noted that even though one element appears in a set repeatedly, it is considered same as appearing once. A = B if and only if A ⊆ and B and B ⊆ A This information is extremely important to prove the equality of sets.
Proper Subset
A is the proper subset of B if and only if A ⊆ B and A ≠ B That means, every element of A is also an element of B and B has at least one element which is not an element of A. For example, A = {1, 2} , B = {1, 2, 3} . To express A as a proper subset of B, A ⊂ B is written.
1) For any set A ,A ⊆ A. This is because x ∈ A ⇒ x ⊆ A
2) For any set A , Ø ⊆ A . This is because if emptyset subseteq A do not happen, Ø will have one element ≈ that A will not have. But this is never true because Ø is an empty set. So , emptyset subseteq A It should be mentioned that empty set or Ø is a proper subset of any set.
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