## Glossary

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# Sets and functionsSubsets

The idea of set equality can be broken down into two separate relations: two sets are equal if the first set contains all the elements of , and .

Definition (Subset)
Suppose and are sets. If every element of is also an element of , then we say is a subset of , denoted .

If we visualize a set as a potato and its elements as dots in the blob, then the subset relationship looks like this:

Here has elements, and has elements.

Two sets are equal if .

The relationship between "" and "=" has a real-number analogue: we can say that x=y if and only if .

Exercise
Think of four pairs of real-world sets which satisfy a subset relationship. For example, the set of cars is a subset of the set of vehicles.

Exercise
Suppose that is the set of even positive integers and that is the set of positive integers which are one more than an odd integer. Then .

Solution. We have , since the statement " is a positive even integer" the statement " is one more than an odd number". In other words, implies that .

Likewise, we have , because " is one more than an positive odd integer" " is a positive even integer".

Finally, we have , since .

Exercise
Drag the items below to put the sets in order so that each set is a subset of the one below it.

Bruno