Subsets

In mathematics, a subset is a concept that involves a set of elements, all of which belong to another, larger set. This larger set is commonly referred to as the 'parent set'. This is a fundamental part of set theory, and to better understand it, let's look at a practical example. $$ A \subseteq B \wedge A \ne B $$

Example

Consider a set A that comprises of these elements: 1, 2, 3, and 4.

Written in mathematical notation, it appears like this:

$$ A = \{ 1,2,3,4 \} $$

Next, let's introduce set B = {1, 2}. This set, B, is deemed a 'subset' of A since all elements found in B also exist in A.

$$ B = \{ 1,2 \} $$

In this context, set A takes the role of the parent set, and set B becomes the subset.

Should you visualize these two sets with Venn diagrams, you'd observe that the area representing subset B falls wholly within the larger area representing the parent set, A.

The relationship established between a set and its subset is mathematically represented by the symbol B ⊆ A.

This implies that set B is wholly contained within set A.

$$ B \subseteq A $$

Alternatively, you could express this relationship as A ⊇ B, signifying that set A encompasses set B.

This representation offers a different perspective but communicates the same information.

You can employ these mathematical symbols to establish and illustrate relationships among several sets.

For instance, if A ⊇ B and B ⊇ C, you can deduce that A ⊇ C.

$$ A \supseteq B \supseteq C $$

This logically means that if B is a subset of A and C is a subset of B, C must also be a subset of A.

Strict Inclusion

Let's shift gears and talk about 'Strict Inclusion'.

This term describes a unique relationship between two sets.

Suppose we have a set A that is strictly included in another set B. This would mean all elements of A are also elements of B, but critically, A is not identical to B. $$ A \subseteq B \wedge A \ne B $$

We signify this relationship with the symbol ⊂.

Let's apply this to a practical example.

We'll consider two sets, A and B:

$$ A = \{ 1,2, 3, 4 \} $$

$$ B = \{ 1,2 \} $$

In this scenario, set B is strictly included in A as all elements of B are also in A, but the two sets, A and B, are not identical.

$$ B \subset A $$

Here, B is said to be a 'proper subset' of A. It's interesting to note that in this specific case, both forms of inclusion apply.

In this particular case, both forms of inclusion hold true. $$ B \subset A $$ $$ B \subseteq A $$

However, consider two identical sets: $$ A = \{ 1,2, 3, 4 \} $$ $$ C = \{ 1,2,3,4 \} $$ In this case, you can only declare that C is a subset of A since all elements of C also exist within set A. $$ C \subseteq A $$ But, since both sets are identical, you cannot assert that C is a proper subset of A. $$ C \require{cancel} \cancel{ \subset } A $$ This brings us to a noteworthy point about 'Improper Subsets'.

Improper Subsets

All sets possess at least two subsets that we classify as 'improper subsets':

• The Empty Set
  The empty set, symbolized as Ø, is an improper subset of any other set. By definition, a set is regarded as an empty set if it does not contain any elements. $$ Ø \subseteq A $$

  For instance, the empty set Ø={ } is an improper subset of the set A, i.e., Ø⊆A. $$ Ø \subseteq \{1, 2, 3, 4 \} $$

• The Identical Set
  The second form of improper subset is the 'identical set'. As the name suggests, every set is considered an improper subset of itself. $$ A \subseteq A $$

  For example, the set A={1,2,3,4} has itself as an improper subset, i.e., A⊆A. $$ \{ 1,2,3,4 \} \subseteq \{ 1,2,3,4 \} $$

Through these examples and explanations, we hope to provide a clear understanding of subsets, their relationships with parent sets, and the distinctions between proper and improper subsets.

In some mathematics books, only the set that is equal to itself is considered an improper subset, while the empty set is not, depending on the definition of 'proper subset' being used.