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What are sets in mathematics?
A set in mathematics is simply an organized collection of well-defined objects or elements. In other words, a set refers to the collection of elements that are unique. It can have any group of collected items such as type of automobiles, days of a week, letter of alphabets, etc. The representation of a set is done by a capital alphabetic letter.
- Elements of a set:
Assume a set A = {7,8,9}. In this A is the set and 7, 8, 9 are the elements or members of a set. The elements in a set can’t be repeated but can be in any sequence and are enclosed in a curly bracket. When an element is a part of a set then it is denoted by ‘∈’ and when an element is not part of a set then it is denoted by ‘∉’.
- The cardinality of a set:
The cardinal number, order of a set, or cardinality of a set implies a total number of elements in a set. The sets are a collection of unique elements and all elements in a set are related to each other.
How a set can be represented?
- Semantic form: This form is used to show what are the elements of a set. For Example, Set B has the first four even natural numbers.
- Roster Form: Most common way to represent the set is curly brackets {}. Here elements are separated by commas and their order doesn’t matter. For Example, Set A = {4,5,6,7}.
- Set Builder Form: In this form, a vertical bar is used in its representation, with a text that describes the character of elements of a set. For example, B = {k| k is an odd number, k ≤ 10}.
What are the set formulas?
Following are the most important set formulas:
For any two given sets, say A and B,
- n(A U B) = n(A) + n(B) – n(A ∩ B)
- n (A ∩ B) = n(A) + n(B) – n(A U B)
- n(A) = n(A U B) + n(A ∩ B) – n(B)
- n(B) = n(A U B) + n(A ∩ B) – n(A)
- n(A – B) = n(A U B) – n(B)
- n(A – B) = n(A) – n(A ∩ B)
For any two disjointed sets A and B,
- n(A U B) = n(A) + n(B)
- A ∩ B = ∅
- n(A – B)= n(A)
- Operations on a set:
- Union of sets: It is denoted as A ∪ B, a set that contains all elements of two sets set A and set B. For example, Set A = {1,2,3} and Set B = {4,5,6}, then A ∪ B = {1,2,3,4,5,6}.
- Intersection of sets: It is denoted as A ∩ B, a set that contains all common elements of two sets set A and set B. For example, Set A = {1,2,3} and Set B = {3,4,5}, then A ∩ B = {3}.
- Complements of sets: The complement of any set, say B, is the set of all elements that are not in set B. It is represented by B’.
- Cartesian Product of sets: Cartesian product of sets, say set A and set B is a set of all ordered pairs (a,b), where, element of set A is a and, the element of set B is b. For example, set A = {1,2} and set B = {chair, table}, then A × B = {(1×chair),(1×table),(2×chair)(2×table)}.
- Difference of sets: Set A difference Set B is a set that doesn’t have elements of B but A. For example, A = {1,2,3} and B = {2,3,4}, then A – B = {1}.
There are different types of sets including several formulas and operations. For a better understanding, we have discussed the basics of the set in this article. Facing difficulty in understanding what set is? Just go to the Cuemath website or read the below-mentioned article.