Sets - Definitions,types and examples of sets

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Today, we’re diving into the fascinating world of sets. Stay tuned as we embark on this interesting adventure!

So, What Are Sets?

In mathematics, sets are collections of well-defined objects. They are usually denoted by capital letters, and their elements are enclosed in curly brackets { }. For example, a set A can be defined as the collection of even numbers less than 10.This statement can be written as A = {2, 4, 6, 8}.

Members/Elements of Sets

The members or elements of a set refer to the individual objects within that set. For instance, in the set A = {1, 2}, the elements are 1 and 2. The symbol ∈ indicates membership, meaning "is an element of." For example, 1 ∈ A means "1 is a member of set A."

The number of elements in set A can be expressed as n(A) = 2, indicating that set A has two elements.

Let's consider another example. For set H = {5, 10, 15, 20}, we find that n(H) = 4 since there are four elements in set H.

Types of Sets

1. Finite Set: This type has a known last element. For example, set P = {1, 2, 3, 4} and set Q = {1, 2, 3, 4, ..., 100} are finite sets since their last members are defined or known.

Note: If a set has ellipses (...) after its last member, it is an infinite set. Since there are no ellipses after 100 in set Q, it is finite.

2. Infinite Set: This type has no known last member. For example, set R = {1, 2, 3, 4, ...} is infinite as its last member is unknown.

3. Equal Sets: These have same members. For instance, set E = {1, 3, 5} and set F = {5, 1, 3} are equal sets because they contain the same elements.

4. Equivalent Sets: These have the same number of members. For example, set A = {1, 2, 3, 4} and set B = {5, 6, 7, 8} are equivalent since both have four elements.

5. Universal Set: This set contains all elements relevant to a particular discussion, without repeating any. For instance, if set A = {1, 2, 3} and set B = {3, 4, 5}, the universal set U = {1, 2, 3, 4, 5}.

6. Subset: A subset consists of members that are also in the universal set. For example, if U = {1, 2, 3, 4, 5}, sets A = {1, 2, 3} and B = {3, 4, 5} are both subsets of U.

Union and Intersection of Sets

Let’s consider set M = {1, 2, 4} and set N = {2, 3, 5}. The intersection, M ∩ N = {2}, represents the elements common to both sets.

The union of the sets combines their elements without repetition. For M and N, the union M ∪ N = {1, 2, 3, 4, 5}. Here, we only list 2 once, even though it appears in both sets.

Disjoint Sets

Disjoint sets have no elements in common. For instance, if set A = {1, 2, 3} and set B = {4, 5, 6}, then A and B are disjoint sets since A ∩ B = {}.

Assessment Test

Given set P = {2, 3, 5, 7, 11} and set Q = {1, 3, 5, 7, 9}, determine whether the following statements are true or false:

i. n(P) = n(Q)

ii. 2 ∉ Q

iii. n(P ∪ Q) = 6

iv. P ⊂ Q

Feel free to share your answer in the comments! You can also reach out for solutions.