Determine which of the following statements are true Given a

Determine which of the following statements are true. Given a short proof, justification, or counterexample for each.

a) The empty set is a subset of every set.

Solution

Part a) True statement.

Explaination : Set A is a subset of set B if each and every element of set A is also an element of set B. And if A is an empty set, then it means it has no elements, and therefore all of its elements (which in fact are none) belong to set B no matter what set B we are dealing with. So, an empty set is a subset of every set.

Part b) False.

Explaination : Counter example is :

Let A = { (1,2) , (5,4) , (6,7), (2,9) }

B = {(1,2) , (2,9) }

So, B is a subset of A, but A does not equal B.

Part c) True Statement

Explaination : Every set is a subset of itself. So, B is a subset of B. And A is a subset of A. And if B=A, then B is a subset of A.

I hope this makes sense. :-)