Subsets

If all of the items of set 1 are present in set 2, we may say that set 1 is a subset of set 2 and vice versa. Sets are well-defined collections of numbers, alphabets, objects, or any other type of thing as we all know. If set 1 consists of the letters A, B, and C, and set 2 consists of the letters A, B, C, D, E, and F, we may say that set 1 is a subset of set 2 since all of the items in set 1 are also present in set 2.

Let us learn about subsets, as well as their kinds (proper subset and improper subset), through a variety of illustrations.

Definition of subsets

When all of the elements of one set A are contained within another set B, the set A is said to be a subset of the other set. In this particular instance, we say

A is a subset of the set B. (or)

B is a superset of A in terms of its elements.

If A is the set of natural numbers and B is the set of all whole numbers, then A is a subset of B since all natural numbers are present in the set of all whole numbers). To put it another way, we may say that

A = a collection of natural numbers (e.g., 1, 2, 3,…)

B = A set of whole numbers consisting of the digits 0, 1, 2, 3,…

Because every element of A is present in B, A B.

Subsets

Proper Subset

A proper subset of a set is any subset of the set that is not the set itself. We all know that every set is a subset of itself, but we don’t realise that every subset of itself is a correct subset of itself. Suppose A = {1, 2, 3}, and its appropriate subsets are 1, 2, 3, and 3, but the set itself is not one of these suitable subsets because it does not include the elements 1, 2, 3, and 3.

Improper Subset

An improper subset is a subset of a set that is NOT a legitimate subset of the set in question. In other words, every set A has exactly one inappropriate subset, which happens to be the set A itself. Some instances of incorrect subsets are shown below.

The improper subset of 1, 2, 3 is the only one that exists.

There is only one inappropriate subset of the pair a, b.

Examples of subsets

The sole criteria for a set A to be a subset of a set B is that every element of A must also be present in both sets B. As a result, below are a few instances of subsets based on this.

A ={ 1, 2, 3} is a subset of A = {1, 2, 3}. B ={ 1, 2, 3, 4, 10}, and so on.

A = p, q, and r represents a subset of B is a collection of all alphabets.

A = set of all even numbers is a subset of A = set of all odd numbers. B is a collection of all integers.

Make a note of the fact that every set is a subset of itself, and that the empty set () is likewise a subset of all other sets.

Conclusion 

In mathematics, a set A is a subset of a set B if all of its constituents are also elements of B; B is thus a A and B can be equal; if they are unequal, then A is a legitimate subset of B. Inclusion refers to the connection of one set being a subset of another (or sometimes containment).