# What is the subset for 2 3

## Subset

In this chapter we look at what a subset is.
Basic knowledge of set theory is assumed to be known.

Repetition

When considering quantities, we are often interested in how they relate to one another. So the question is: What is the relationship between \ (A \) and \ (B \)?

Quantity relationships

• \ (A \) and \ (B \) are the same
• \ (A \) is contained in \ (B \) (or: \ (B \) is contained in \ (A \))
• \ (A \) partially covers \ (B \)
• \ (A \) and \ (B \) are different from each other

In the following we look at two examples for the case that \ (A \) is contained in \ (B \):

example 1

\ (A = \ {{\ color {green} 4}, {\ color {green} 5} \} \)
\ (B = \ {1, 2, 3, {\ color {green} 4}, {\ color {green} 5} \} \)

observation
\ (A \) is contained in \ (B \).

Spelling and speaking style
\ (\ definecolor {naranja} {RGB} {255,128,0} {\ color {naranja} A \ subseteq B} \) (read: "A is a subset of B")

Example 2

\ (A = \ {{\ color {green} 1}, {\ color {green} 2}, {\ color {green} 3}, {\ color {green} 4}, {\ color {green} 5} \} \)
\ (B = \ {{\ color {green} 1}, {\ color {green} 2}, {\ color {green} 3}, {\ color {green} 4}, {\ color {green} 5} \} \)

Observation 1
\ (A \) is contained in \ (B \).

Spelling and speaking 1
\ (\ definecolor {naranja} {RGB} {255,128,0} {\ color {naranja} A \ subseteq B} \) (read: "A is a subset of B")

Observation 2
\ (A \) and \ (B \) are the same.

Spelling and speaking style 2
\ (A = B \) (say: "A equals B")

As the two introductory examples have shown, the following obviously applies:

The subset relation \ (A \ subseteq B \) (or: \ (B \ subseteq A \))
includes the case of set equality \ (A = B \).

annotation

The term “real subset” excludes the equality of sets.

### Definition of a subset

A set \ (A \) is called Subset a set \ (B \),
if every element of \ (A \) also belongs to the set \ (B \):

\ (A \ subseteq B ~ \ Leftrightarrow ~ \ forall x ~ (x \ in A \ Rightarrow x \ in B) \)

The above formula translates as:

\(
\ underbrace {\ vphantom {\ big \ vert} A \ subseteq B} _ \ text {A is a subset of B} ~~
\ underbrace {\ vphantom {\ big \ vert} \ Leftrightarrow} _ \ text {if and only if} ~~
\ underbrace {\ vphantom {\ big \ vert} \ forall x} _ \ text {for all x applies:} ~~
(
\ underbrace {\ vphantom {\ big \ vert} x \ in A \ Rightarrow x \ in B} _ \ text {from x is an element of A follows x is an element of B} ~~
)
\)

### Check quantities for partial quantity relationship

Examples

• Is \ (A = \ {1 \} \) a subset of \ (B = \ {1, 2, 3 \} \)?

solution
\ (A \ subseteq B \) ("A is a subset of B")

Explanation
Every element of \ (A \) is also contained in \ (B \).

• Is \ (A = \ {4 \} \) a subset of \ (B = \ {1, 2, 3 \} \)?

solution
\ (A \ nsubseteq B \) ("A is not a subset of B")

Explanation
The element "4" is not contained in \ (B \).