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Sets and functionsSubsets

Czas czytania: ~15 min

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 S and T are sets. If every element of T is also an element of S, then we say T is a subset of S, denoted T \subset S.

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

Here S has elements, and T 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 E is the set of even positive integers and that F is the set of positive integers which are one more than an odd integer. Then .

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

Likewise, we have F \subset E, because "n is one more than an positive odd integer" "n is a positive even integer".

Finally, we have E = F, since .

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

\mathbb{R}
\mathbb{N}
\mathbb{Q}
\mathbb{Z}
Bruno
Bruno Bruno