SUBSETS AND VENN DIAGRAMS

**Representing subsets on a Venn diagram**

When we know that *S* is a subset of *T*, we place the circle representing *S* inside the circle representing *T*. For example, let *S*={0,1,2}, and *T*={0,1,2,3,4}. Then *S* is a subset of *T*, as illustrated in the Venn diagram below.

Make sure that 5, 6, 7, 8, 9 and 10 are placed outside both circles.

⛲ Example 0: **Subsets**

If *A*={a,b,c} then *A* has eight different subsets:

Notice that

*A*⊆

*A*and in fact each set is a subset of itself. The empty set Ø is a subset of every set.

⛲ Ex1. Write down the subsets of the following sets.

(i) {1,2,3} (ii) Ø

✍

(i) The subsets of {1,2,3} are

(ii) Clearly, {Ø} is the power set of empty set Ø. Now, its subsets are Ø and {Ø}.

⛲ Ex2. Write down all the subsets of the following sets:

(i) {a}

(ii) {a,b}

(iii) {1,2,3}

(iv) Ø

✍

(i) The subsets of {a} are Ø and {a}.

(ii) The subsets of {a, b} are Ø, {a}, {b}, and {a, b}.

(iii) The subsets of {1,2,3} are Ø, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}

(iv) The only subset of Ø is Ø.

⛲ Ex3. List all the subsets of the set {-1,0,1}.

✍ Solution:

Let *A*={-1,0,1}. The subset of *A* having no element is the empty set Ø. The subsets of *A* having one element are {-1}, {0}, {l}. The subsets of *A* having two elements are {-1,0}, {-1,1}, {0,1}. The subset of *A* having three elements of *A* is *A* itself. So, all the subsets of *A* are (1), {-1}, {0}, {1}, {-1,0}, {-1,1}, {0,1} and {-1,0,1}.

📌 Exercises (solved)

A. List all the subsets of the following sets.

1. {1,2,3,4}. The subsets of {1,2,3,4} are:

{}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}.

2. {{ℝ}}. The subsets of {{ℝ}} are: {} and {{ℝ}}.

3. {Ø}. The subsets of {Ø} are {} and {Ø}.

4. {ℝ,{ℚ,ℕ}. The subsets of {ℝ,{ℚ,ℕ} are {}, {ℝ}, {{ℚ,ℕ}}, {ℝ,{ℚ,ℕ}}.

B. Write out the following sets by listing their elements between braces.

1. {*X*∶*X*⊆{3,2,a} and |*X*|=2}={{3,2},{3,a},{2,a}}

2. {*X*∶*X*⊆{3,2,a} and |*X*|=4}={}=Ø

⛲ Example 5: **Distinct Subsets**

a) Determine the number of distinct subsets for the set {S,L,E,D}.

b) List all the distinct subsets for the set {S,L,E,D}.

c) How many of the distinct subsets are **proper subsets**?

✍ Solution:

a) Since the number of elements in the set is 4, the number of distinct subsets is 2=2^{4}=2⋅2⋅2⋅2=16.

b)

c) There are 15

**proper subsets**. Every subset except {S,L,E,D} is a

**proper subset**

This idea of “making” a subset can help us list out all the subsets of a given set *B*. As an example, let *B*={*a*,b,c}. Let’s list all of its subsets.

One way of approaching this is to make a tree-like structure. Begin with the subset {}, which is shown on the left of Figure 2. Considering the element *a* of *B*, we have a choice: insert it or not. The lines from {} point to what we get depending whether or not we insert *a*, either {} or {*a*}. Now move on to the element b of *B*. For each of the sets just formed we can either insert or not insert b, and the lines on the diagram point to the resulting sets {}, {b}, {*a*}, or {*a*,b}. Finally, to each of these sets, we can either insert c or not insert it, and this gives us, on the far right-hand column, the sets {}, {c}, {b}, {b,c}, {*a*}, {*a*,c}, {*a*,b} and {*a*,b,c}. These are the eight subsets of *B*={*a*,b,c}.

We can see from the way this tree branches out that if it happened that *B*={*a*}, then *B* would have just two subsets, those in the second column of the diagram. If it happened that *B*={*a*,b}, then *B* would have four subsets, those listed in the third column, and so on. At each branching of the tree, the number of subsets doubles. Thus in general, if |*B*|=*n*, then *B* must have 2^{n} subsets.

[__Fact__] If a finite set has n elements, then it has 2^{n} subsets.

For a slightly more complex example, consider listing the subsets of *B*={1,2,{1,3}}. This *B* has just three elements: 1, 2 and {1,3}. At this point you probably don’t even have to draw a tree to list out B’s subsets.

You just make all the possible selections from *B* and put them between braces to get

These are the eight subsets of

*B*. Exercises like this help you identify what is and isn’t a subset. You know immediately that a set such as {1,3} is

__not__a subset of

*B*because it can’t be made by selecting elements from

*B*, as the 3 is not an element of

*B*and thus is not a valid selection. Notice that although {1,3}⊄B (read: {1,3}

**isn’t a proper-subset of**B), it is true that {1,3}∈B. Also, {{1,3}}⊂

*B*(read: {1,3}

**is a proper-subset of**B).