Question

Without using the equality symbol = or the subset symbol ⊆ (or their negations), just using logical notation introduced in class, express the statement that for given sets A, B, and C : either A = B , or B ⊆ A , or, for some x ∈ B, x ∉ C

Answer 1

Let us assume that there exists a elment x which belongs to the set B

Now for every element belonging to B, that element also belong to A

The above statement implies that

Now there can be only two possibilities either they have same elments (all elements are same in both A and B) or there are some elements which belong to A but doesn't belong to B

Case 1: Let us assume the first case  they have same elments (all elements are same in both A and B)

In this case, we can write

and similarly since the number of elements are same, so we can write the inverse of this statement which implies

The above two statements yield

Case 2: Let us assume the first case  there are some elements which belong to A but doesn't belong to B

In this case, we can write

and now since we now there are some elements x such that which belong to A but not to B

The above two statements yield