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# Subsets

A set $A$ is a subset of $B$ if all the elements of $A$ belong to $B$. We write this $A\subseteq B$.

The set of mammals is a subset of the set of all animals.

The set of the natural numbers $\N$ is a subset of the integers $\Z$. But the set $\Q$ of the irrational numbers is not.

The set of vowels is a subset of the set of all letters.

$A$ is a proper subset if it is a subset different from $B$. It contains some elements of $B$ but not all of them. We write this as $A\subset B$.

$$\N\subset\Z,\;\Z\subseteq\Z,\;\{\Tred{A},\Tred{I}\}\subset\{\Tred{A},B,\Tred{I},L\}$$

The subset of a subset is a subset.

$$\N\subset\Z\subset\Q\subset\R$$
The set of primary colours and the set of secondary colours are both subsets of the set of all colours. They have no elements in common.