Transcribed Image Text: Definition of subset and empty set. Let A and S be sets. Then A is a subset of S, written A C S, if every element in A is also in S. The empty set is the set containing no elements and is denoted Ø. (a) List all subsets of two elem

Transcribed Image Text: Definition of subset and empty set. Let A and S be sets. Then
A is a subset of S, written A C S, if every element in A is also in
S. The empty set is the set containing no elements and is denoted Ø.
(a) List all subsets of two elements that may be identified within the
set {a, b, c, d}, and say how many such subsets there are. (b) Is it true
that ØC {a,c}

In set theory, a subset is a set that contains elements from another set. More formally, if we have two sets A and S, we say that A is a subset of S if every element in A is also an element in S. This is denoted as A ⊆ S.

On the other hand, the empty set is a set that contains no elements. It is commonly denoted as Ø or {}. It is important to note that the empty set is a subset of every set.

Now, let’s move on to the specific questions:

(a) List all subsets of two elements that may be identified within the set {a, b, c, d}, and say how many such subsets there are.

In this question, we are asked to find all possible subsets of two elements within the set {a, b, c, d}. To do this, we can consider all possible combinations of two elements from the given set. In this case, the elements are: a, b, c, and d.

The subsets of two elements are:

{a, b}, {a, c}, {a, d},
{b, c}, {b, d},
{c, d}.

There are a total of 6 subsets of two elements that can be identified within the set {a, b, c, d}.

(b) Is it true that Ø ⊆ {a, c}?

The empty set, Ø, is a subset of every set. Therefore, it is true that Ø ⊆ {a, c}.

To prove this, we need to show that every element in the empty set is also in the set {a, c}. However, since the empty set contains no elements, there are no elements to check. Thus, the condition for being a subset is automatically satisfied, making Ø a subset of {a, c}.

In conclusion, every set is considered a subset of itself, and the empty set is a subset of every set. Additionally, in the given set {a, b, c, d}, there are 6 subsets of two elements. Finally, it is true that Ø is a subset of {a, c}.