If X and Y are sets and every element of X is also an element of Y, then we say or write:
Table of contents |
2 Examples 3 Simple results |
PROPOSITION 1: Given any three sets A, B and C, if A is a subset of B and B is a subset of C, then A is a subset of C.
PROPOSITION 2: Two sets A and B are equal if and only if A is a subset of B and B is a subset of A.
PROPOSITION 3: The empty set is a subset of every set.
Proof: Given any set A, we wish to prove that {} is a subset of A. This involves showing that all elements of {} are elements of A. But there are no elements of {}.
For the experienced mathematician, the inference "{} has no elements, so all elements of {} are elements of A" is immediate, but it may be more troublesome for the beginner. Since {} has no members at all, how can "they" be members of anything else? It may help to think of it the other way around. In order to prove that {} was not a subset of A, we would have to find an element of {} which was not also an element of A. Since there are no elements of {}, this is impossible and hence {} is indeed a subset of A.
These propositions show that ⊆ is a partial order on the class of all sets, and {} is a bottom element.