prove that if A ⊆ B, B ⊆ C, and C ⊆ A, then A = C.

Question

foundation of mathematics 

Answer 1

Let  x be any element of A.

Then x is an element of B since A is a subset of B.

Then x is an element of C since B is a subset of C.

So A is a subset of C because any element, x, of A is also in C.

 

Since C is a subset of A, then any element, y,

of C is also in A.

 

Two sets that are subsets of each other must be the same.

 

end of proof