Subject: sets
Name: narayana
Who are you: Teacher

how to show a non empty set is a subset of every set


Hi Narayana.

Are you sure you wrote that correctly? What you wrote is false. Perhaps you meant that you wanted to show that the empty set is a subset of every set.

If so, there are probably many ways of convincing yourself (or students) that this is the case.

  1. The set A is a subset of the set B if and only if every element of A is also an element of B. If A is the empty set then A has no elements and so all of its elements (there are none) belong to B no matter what set B we are dealing with. That is, the empty set is a subset of every set.

  2. Another way of understanding it is to look at intersections. The intersection of two sets is a subset of each of the original sets. So if {} is the empty set and A is any set then {} intersect A is {} which means {} is a subset of A and {} is a subset of {}.

  3. You can prove it by contradiction. Let's say that you have the empty set {} and a set A. Based on the definition, {} is a subset of A unless there is some element in {} that is not in A. So if {} is not a subset of A then there is an element in {}. But {} has no elements and hence this is a contradiction, so the set {} must be a subset of A.

An example with an empty set and a non-empty set might be this: the (set of all women who have walked on the moon) and the (set of all astronauts). Examine the three arguments above with this example in mind

We hope this helps,
Stephen and Penny