Just starting set theory (privately) and already a difficulty. A is a subset of B if all the elements of A are also in B. The empty set has no elements. But the empty set is a subset of A?? Seems illogical to me or have I missed something?
Just starting set theory (privately) and already a difficulty. A is a subset of B if all the elements of A are also in B. The empty set has no elements. But the empty set is a subset of A?? Seems illogical to me or have I missed something?
A is a subset of B iff for all elements a of A a is in B.
(Assume there exists a in the empty set, now a false proposition implies anything
so this assumption implies that a is in B, so for all a in the null set a is in B)
Also A is not a subset of B if there exists an element a of A which is not in B.
Now this is always true of the empty set as there are no elements of A which are not in B.
So the empty set is a subset of every set.
RonL
I read the line in your post which I've underlined above as " Now this is always true of the empty set as there are no elements of the empty set that are not in A".
A true statement since the empty set has no elements .. period. This same argument would apply to all sets.
Thanks. I think I've got it now (although I must admit that the initial clause before the "Also A" about a false proposition implying anything, is still making me wonder about its significance.