The set theory by Sabrina Di guida

What is a set

Sabnyanya — 2018-11-30 09:47:31 UTC

A set is a collection of objects that are called elements of the set

A set is usualy denoted by capital letters,A,B,C... and the elements are denoted by samall letters, a,b,c,d...

Sabnyanya — 2018-11-30 15:01:22 UTC

THERE ARE THREE WAYS TO RAPRESENT A SET

Sabnyanya — 2018-11-30 15:14:18 UTC

1.TABULAR FORM

Sabnyanya — 2018-11-30 15:18:12 UTC

A={1,2,3,4,5};B={1,3,5,7,9...}...

2. DESCRIPTIVE FORM

Sabnyanya — 2018-11-30 15:23:50 UTC

A= Set of frist five natural numbers.
B= Set of positive odd integers less than or equal to fifty.

3. SET BUILDER FOR

Sabnyanya — 2018-11-30 15:31:01 UTC

A={x:x€N^x<_5}
B={x:x€E^0

EMPTY OR NULL SET

Sabnyanya — 2018-12-07 09:43:10 UTC

The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set

Example:

Sabnyanya — 2019-01-24 17:02:19 UTC

1. Odd numbers which are divisible by 2 are well defined objects for an empty set.
2. The positive integers which are less than -1 are also well defined for en empty set.

Finite set

Sabnyanya — 2019-01-24 17:09:57 UTC

A set is said to be a finite set if it is either void set or the process of counting of elements surely comes to an end is called a finite set. In a finite set the element can be listed if it has a limited i.e. countable by natural number 1, 2, 3, ……… and the process of listing terminates at a certain natural number N.

Sabnyanya — 2019-01-24 17:17:54 UTC

The number of distinct elements counted in a finite set S is denoted by n(S). The number of elements of a finite set A is called the order or cardinal number of a set A and is symbolically denoted by n(A).

Sabnyanya — 2019-01-24 17:19:00 UTC

The element does not occur more than once in a set.

Example:

Sabnyanya — 2019-01-24 17:19:28 UTC

1. Let P = {5, 10, 15, 20, 25, 30}

Then, P is a finite set and n(P) = 6.

Infinite set

Sabnyanya — 2019-01-24 17:20:55 UTC

A set is said to be an infinite set whose elements cannot be listed if it has an unlimited (i.e. uncountable) by the natural number 1, 2, 3, 4, ………… n, for any natural number n is called a infinite set.

Example:

Sabnyanya — 2019-01-24 17:21:55 UTC

Set of all points in a line segment is an infinite set.

. N = {1, 2, 3, ……….} i.e. set of all natural numbers is an infinite set.

Example:

Sabnyanya — 2019-01-24 17:26:11 UTC

A={2, 3, 4, 5, 6, 7, 8} is proper subset of B={1, 2, 3, 4, 5, 6, 7, 8}.

If A is a subset of B, then B is called the super set of A. The symbol ⊂ is called the inclusion symbol. If A is not a subset of B, we write A⊄B. By the definition of a subset, it is clear that the empty set and the set A itself are always subsets of A. These two subsets are called the improper subsets of A.

Equal and equivalent set

Sabnyanya — 2019-01-24 17:27:36 UTC

Two sets can either be equivalent, equal or unequal to each other.

Sabnyanya — 2019-01-24 17:30:39 UTC

Two sets A and B can be equal only if each element of set A is also the element of the set B. Also, if two sets are the subsets of each other, they are said to be equal. This is represented by:

A = B

A ⊂ B and B ⊂ A ⟺ A = B

Sabnyanya — 2019-01-24 17:32:23 UTC

If the condition discussed above is not met, then the sets are said to be unequal. This is represented by

A ≠ B

Equivalent set

Sabnyanya — 2019-01-24 17:40:57 UTC

To be equivalent, the sets should have the same cardinality. This means that there should be one to one correspondence between elements of both the sets. Here, one to one correspondence means that for each element in the set A, there exists an element in the set B till the sets get exhausted.

Sabnyanya — 2019-01-24 17:43:39 UTC

Definition 1: If two sets A and B have same cardinality, if there exists a objective function from set A to B.

Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B).

Set Operations

AliceWJM — 2019-01-24 17:51:39 UTC

The difference between two sets

Sabnyanya — 2019-01-24 17:53:39 UTC

We say the difference between two sets A and B, considered in the order,the set of elements of A that do not belong to B.

Union

AliceWJM — 2019-01-24 17:54:51 UTC

The symbol ∪ is employed to denote the union of two sets. The union of two sets is the set containing all of the elements from both of those sets. The set A ∪ B you read “A union B” or “the union of A and B”.

A ∪ B = {x : x ∈ A or x ∈ B}

Sabnyanya — 2019-01-24 17:58:56 UTC

We write A-B and read A minus B.
In symbols A-B {x|x A and x ₵ B}.

Intersection

AliceWJM — 2019-01-24 18:00:11 UTC

The intersection operation is denoted by the symbol ∩.
The set A ∩ B you read “A intersection B” or “the intersection of A and B” is defined as the set composed of all elements that belong to both A and B.

A ∩ B = {x : x ∈ A and x ∈ B}

Disjoint Sets

AliceWJM — 2019-01-24 18:05:52 UTC

If E denotes the set of all positive even numbers and O denotes the set of all positive odd numbers, then their union yields the entire set of positive integers, and their intersection is the empty set. Any two sets whose intersection is the empty set are said to be disjoint.

Sabnyanya — 2019-01-24 18:11:15 UTC

AliceWJM — 2019-01-24 18:14:42 UTC

Complementary set of a set

Sabnyanya — 2019-01-24 18:15:09 UTC

In set theory, the complement of a set A refers to elements not in A.

When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A.

The relative complement of A with respect to a set B, also termed the difference of sets A and B, written B ∖ A, is the set of elements in B but not in A.

Sabnyanya — 2019-01-24 18:18:40 UTC

Cartesian Product

AliceWJM — 2019-01-24 18:21:11 UTC

The Cartesian product A × B between two sets A and B is the set of all possible ordered pairs with first element from A and second element from B. For example, if A = {x, y} and B = {3, 6, 9},
then A × B = {(x, 3), (x, 6), (x, 9), (y, 3), (y, 6), (y, 9)}.

A×B={(x,y):x∈A and y∈B}.

Power set and the partition of a set

Sabnyanya — 2019-01-24 18:43:35 UTC

Partition of a set

Sabnyanya — 2019-01-24 18:46:53 UTC

A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets[2] (i.e., X is a disjoint union of the subsets).

Equivalently, a set P is a partition of X if, and only if, it does not contain the empty set and:

The union of the elements of P is equal to X. (The elements of P are said to cover X.)
The intersection of any two distinct elements of P is empty. (We say the elements of P are pairwise disjoint.)

AliceWJM — 2019-01-24 18:47:12 UTC

Power Set

AliceWJM — 2019-01-24 18:57:33 UTC

the power set of any set A is the set of all subsets of A, including the empty set and A itself, variously denoted as P(A)or P(S

For the set {a,b,c}:

The empty set {} is a subset of {a,b,c}
And these are subsets: {a}, {b} and {c}
And these are also subsets: {a,b}, {a,c} and {b,c}
And {a,b,c} is a subset of {a,b,c}

And altogether we get the Power Set of {a,b,c}:

P(S) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

AliceWJM — 2019-01-24 19:28:48 UTC