# 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 \).

### More about set theory

In connection with set theory, there are some topics that are repeatedly asked about in exams. It is therefore worthwhile to work through the following chapters one after the other.

*Note:*To define mathematical symbols, a colon is usually used in front of an equal sign, with the expression on the left (next to the colon) being defined by the other. The colon equals sign \ (: = \) is spoken "

*is by definition the same*Often the colon is simply left out.

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