Theory of sets Author Topic: Theory of sets  (Read 1725 times) tanzina_diu

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« on: March 31, 2016, 11:43:37 AM »
Once upon a time there was a gorilla. Then there came another gorilla. In a few days there were a lot of gorillas. They made a community. We can say that this community is one set that consists of its members. Gorillas are elements of the set. One intelligent man is not an element of such a set. Look at our Solar system. It consists of the Sun and planets around it. Our Solar system is a set that consists of its members. Earth is one of them.
A set is a collection of objects.
A set is definite if all its members are known. There are a lot of sets: a set of letters, a set of numbers....
letters = {a, b, c, ...x, y z}
numbers = {0, 1, 2, 3, ...}
Three points ... have the same meaning as etc. We use the symbol = and brackets {,} to denote the fact that specific elements belong to a set.
Tanzina Hossain
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« Reply #1 on: March 31, 2016, 11:44:08 AM »
We denote sets with uppercase letters, for example: A, B, C, S, X, Y,…
The elements of sets are denoted by lowercase letters: a, b, c, x, y,….
The relation that an element x belongs to the set S is denoted by x  S (we read it: “x is an element of S”).  If the element y is not a member of the set S, we write y  S (we read it: “y is not an element of S”).
We consider that a set is given if all its members are known. The set named letters has a finite number of elements, hence we call it a finite set. On the other hand, the set named numbers has an infinite number of its elements. Therefore we call it an infinite set.
Some sets can be so long that we can not list all their members. In that case we define a set by giving a property or condition that all elements satisfy.
A = { x | x is each man who thinks he is clever and handsome }
Read this as "A is defined to be the set of x's such that x is each man who thinks he is clever and handsome". Instead of such that it is sometimes better to say for which or where.
The vertical line {..|...} used in this definition is called a constraint bar since only elements that satisfy the condition that follows it may be elements of the set. In fact there is no doubt about it: all elements that satisfy the constraint are members of the set.
A set has no duplicate elements. An element is either a member of a set or not. It cannot be in the set twice.
{1, 2, 3} is the same as the set {1, 3, 2, 3, 1}
Tanzina Hossain
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« Reply #2 on: March 31, 2016, 11:44:40 AM »
Listing an element twice in a set does not add to its membership. The order in which elements are listed does not matter either.
In mathematics there are a lot of statements and definitions that are considered to be true without proof, but there are a lot of statements and theorems that are proved by the use of previous assumptions and definitions.
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« Reply #3 on: April 02, 2016, 01:07:19 PM »
EQUALITY OF SETS
All girls in the class like John. All girls in the class like to eat pizza. The set of girls who like John is equal to the set of girls who like pizza.
Let A and B be sets. Consider that every element of set A is also an element of set B. Set A is equal to the set B if they both have the same members. The equality of sets A and B is denoted by A = B and with (  a  A)( a  A  a  B). This means that each element of the set A is also element of the set B. Here the symbol  means ‘each’. The symbol  means that if a   A, then a  B. We read it "a   A implies a   B" or " from a    A outcomes a  B".
Example: The set {a, b, c, d} is equal to the set {c, a, d, b}, i.e.
{a, b, c, d} = {c, a, d, b}.
The set {a, b, c} is different from the set {a, b}, i.e.
{a, b, c}  {a, b}.
Let N be the set of all natural numbers. If S = {x | x  N, x < 4} and A = {1, 2, 3},
Then S = A.
NULL SET
Let S be the set of Blondies who are clever. The set S is empty. (Remark: This is just a joke, as we know that IQ does not depend on the colour of the hair.)
Let N be the set of all natural numbers. Then the set {x | x  N,   x2 = 6} is an empty set because there is no natural number whose square is 6.
An empty set is a set with no elements. We denote it by the symbol  and call it the null set.
Tanzina Hossain
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« Reply #4 on: April 05, 2016, 04:32:19 PM »
SUBSET
Let A be the set of all Blondies and let B be the set of all women. All the blondies are women, but all women are not blondies. The set of blondies is a subset of the set of women. If all the women were Blondies then the set of Blondies would be equal to the set of women.
A is a subset of  B if every element in set A is also a member of set B. A is a subset of B if a  A implies a  B. We denote that by A  B which is also read 'A is contained in B'.
Definition: Two sets A and B are equal (A = B), if and only if A  B and B  A.
Notification B  A means 'B is superset of A' or 'B contains A'. If A is not a subset of B, we write A  B. That means that there is at least one element in A that is not a member of B.
The null set is a subset of every set.
If A  B and A  B then we call A the proper subset of B and denote it by A  B.
Every set is a subset of itself.
Definition: A is a proper subset of B if:
A is a subset of B
A is not equal to B
i.e. A   B and A  B
Two sets are comparable if one of the sets is a subset of the other set, i.e. A  B or B  A.
Every set is equal to itself, i.e. (A = B)  (B = A) and A = B and B = C implies A = C.
Theorem: If A is a subset of B and B is a subset of C, then A is a subset of C.
A  B and B  C  A  C
Proof: Let x  A. Since A  B  x  B. By the consideration B  C, every element of B, which includes x, is a member of C. We know that x  A  x  C according to the definition A  C.
Tanzina Hossain
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« Reply #5 on: April 05, 2016, 04:32:42 PM »
SETS OF SETS
The objects of a set can be the sets themselves. In that case we are talking about a family of sets or a class of sets, which is denoted by capital letters A, B, C, … In the same time, the capital letters denote their elements.
UNIVERSAL SET
Sometimes we are interested only in the subsets of one set, and other sets have no meaning for our consideration. In such a case we call this set the universal set. The set of all points in one plane can be the universal set.
John is at a party. All the girls at the party are a universal set for John, because he is interested whether they will like him.
POWER SET
The power set is the family of all subsets of a set. If A is a set, then its power set is denoted by 2A or P(A).
For example, if A = {1,2}, then 2A = { {1,2}, {1}, {2},  }.
Tanzina Hossain
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« Reply #6 on: October 26, 2016, 05:31:24 PM » 710001113 