Definition: Given a set A, a subset of A is a set B such that every element in B is also an element in A. //
This definition warrants a few remarks.
The first remark is trivial, and it is that we could have called the set B anything, just as the set A can be any set. The second remark that we should make is that every subset of a set A is itself a set in its own right. Thus, we can ask about subsets of subsets. Namely, if B is a subset of A, then B is also a set, and therefore B will have subsets as well. Note that if C is a subset of B and B is a subset of A, then C is a subset of A also. This is simply due to the fact that since C is a subset of B, every element in C is in B. But since B is a subset of A, every element that is in B is also in A. Since every element in C is in B and every element in B is in A, we have that every element in C is in A.
The last remark is two-fold: every set is a subset of itself, and the empty set (the set with no elements) is a subset of every set. The first part of this remark follows from the fact that if you hand me a set A, then I immediately know that every element in A is in A, and so A is a subset of A (by following the above definition strictly). The second part of this remark follows from the fact that the statement “every element in the empty set is in A” is a vacuous statement, meaning that it is automatically true because the conditions in the statement are not satisfied. Namely, asking about “every element in the empty set” is meaningless, because the empty set has no elements! Thus we can automatically and correctly claim that every element in the empty set is in any set that you hand me, and so the empty set is a subset of every set.
We have now defined what a “substructure” of a set is, and now we can go on to study these structures in a rigorous way. Some sets will have interesting substructure, and others won’t. Had we made a different definition then we’d discover different things, but having made this definition we’re forced to live with whatever logical conclusions it leads us to…
For more on subsets, see lesson 3.