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Subsets (advanced)

$$A$$ is not a subset of $$B$$ if $$A$$ contains an element that is not in $$B$$. $$A$$ and $$B$$ can still have elements in common. We write this $$A\not\subset B$$.

The set of animals that live in water is not a subset of all fish (whales are mammals).

$$$ \Z\not\subseteq\N,\quad\Z\not\subset\Z,\quad \{\Tgreen{A}, \Tred{O}, \Tgreen{I}\}\not\subset\{\Tgreen{A}, B, \Tgreen{I}, L\}$$$

The empty set has no elements. We write it $$\emptyset$$. It is a subset of all sets.

The set of all men with a height of 5m or more is an empty set.

The set of primary colours and the set of secondary colours are both subsets of the set of all colours. They are not a subset of one another.
The set of primary colours and the set of secondary colours are both subsets of the set of all colours. They are not a subset of one another.