Operations on Sets

We learned addition, subtraction, multiplication, and division on numbers. We can call them operations on numbers. We can apply similar concepts to sets, that are operations on sets: union, intersection, and complement.

Before getting into the operations on sets, let us talk about Venn diagrams. I believe that Venn diagrams are powerful and effective way to understand sets.

1. Venn Diagrams

A Venn diagram is a graphical representation

for describing relationships between sets.

Example 1. Let us consider the following example. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Let A = {2, 5, 7} and B = {1, 3, 5, 7, 9}. I will use a Venn diagram to see the relationships among the three sets U, A, and B.

Step 1: Draw a rectangle to represent the universal set U, and label it with U.

Screenshot 2024-07-10 at 10.35.21 AM.png

Step 2: Draw a circle to represent the set A, and label it with A. Additionally, fill the circle with the elements of A.

Step 3: Draw a circle to represent the set B, and label it with B. Fill the circle with elements of B. Since the elements 5 and 7 are common to A and B, do the following.

Step 4: Fill the region outside the two circles with the elements of U which are not contained in A or B, or both.

Question 1. Draw a Venn diagram representing the relationship of the following set. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Let A = {2, 5, 6, 7, 9} and B = {1, 3, 5, 7, 9}.

Answer

Question 2. Draw a Venn diagram representing the relationship of the following set. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Let A = {2, 4, 6, 8} and B = {1, 3, 5, 7, 9}.

Answer

2. Operations on Sets

a. Union

The union of the sets A and B is the set consisting of elements in A or B, or both. It is denoted by A ∪ B.

Let us describe the union of A and B using a Venn diagram. Here, we assume that the universal set U is given.

Step 1: As we did previously, first draw a Venn diagram as follows.

Step 2: Shade the circle representing A because the union A ∪ B contains all the elements of A.

Screenshot 2024-07-12 at 10.11.46 PM.png

Step 3: Shade the circle representing B because the union A ∪ B also contains all the elements of B.

Screenshot 2024-07-12 at 10.12.00 PM.png

As you can see, the purple shaded part is common to A and B. In other words, the elements in the purple part are repeated. As I mentioned in "1:Sets" note, we don't list repeated elements more than once. So, we end up with the following diagram for A ∪ B.

Yellow shaded part = A ∪ B

Screenshot 2024-07-12 at 10.29.42 PM.png

When considering the union of two sets, you don't need to take the universal set into account. You can just focus on the two sets.

Example 2. Find the union of two sets A = {a, b, c, d} and B = {b, d, e, f}. The union of A and B contains all the elements from A and B.

Step 1: Write all the elements of A: A ∪ B = {a, b, c, d, ...... }

Step 2: Write all the elements of B, but if an element of B is already in A, we skip the element: A ∪ B = {a, b, c, d, b, d, e, f}

Thus, the union A ∪ B = {a, b, c, d, e, f}.

Let us use a Venn diagram to find the union A ∪ B.

Screenshot 2024-07-13 at 10.04.37 PM.png

From the above Venn diagram, the yellow shaded part represents A ∪ B.

Thus, A ∪ B = {a, b, c, d, e, f}.

Question 3. Find the union of A and B, A ∪ B.

(a) A = {apple, grape, kiwi} and B = {avocado, orange}

(b) A = {3, 6, 9, 12, 15} and B = {4, 8, 12, 16, 20}

Answer

b. Intersection

The intersection of two sets A and B is the set containing the elements common to both of A and B. It is denoted by A ∩ B.

Let us describe the intersection of A and B using a Venn diagram. Let us assume that the universal set U is given. (Similar to the union, , we don't need to take the universal set in account when considering the intersection.)

Step 1: As we did previously, first draw a Venn diagram as follows.

Step 2: Shade the circles representing A and B, respectively, with different colors.

Step 3: From the above diagram, we take only the purple shaded part for A ∩ B because purple shaded part is in both A and B, simultaneously.

Purple shaded part = A ∩ B

Example 3. Find the intersection of two sets A = {a, b, c, d} and B = {b, d, e, f}. The intersection of A and B contains the elements common to both A and B.

Step 1: Determine which elements of A are also contained in B.

They are b and d.

Step 2: Write the elements we've found in Step 1 as the elements in A ∩ B.

Thus, the intersection A ∩ B = {b, d}.

Let us use a Venn diagram to find the intersection A ∩ B.

First, fill each part with elements in A or B. Then shade the overlapping common part.

From the above Venn diagram, A ∩ B = {b, d}.

Question 4. Find the intersection of A and B, A ∩ B.

(a) A = {apple, grape, kiwi} and B = {avocado, orange}

(b) A = {3, 6, 9, 12, 15} and B = {4, 8, 12, 16, 20}

Answer

c. Complement

Let U be a universal set and A be a set in U (i.e., A is a subset of U).

The complement of A is the set of elements of U which are not in A. It is denoted by A' or .

From now on, I will use A' to represent the complement of A. Let us describe the complement of A, given the universal set U, using a Venn diagram.

Step 1: As we did previously, first draw a Venn diagram as follows.

Step 2: Shade the part outside of the circle representing A. This represents the complement of A, denoted by A'.

Green shaded part = A'

Example 4. Find the complement of A = {b, d, e} given the universal set U = {a, b, c, d, e, f, g, h}. The complement of A contains the elements that are in U but not in A.

Step 1: Cross out the elements of A from the elements in U.

U = {a, b, c, d, e, f, g, h}.

Step 2: Write the remaining elements of U as the elements of A'.

Thus, the complement of A, A' = {a, c, f, g, h}.

We will use a Venn diagram to find the complement of A.

First, fill the circle with elements in A, and fill the part outside of the circle with the remaining elements of U. Then shade the outside part.

Screenshot 2024-08-01 at 10.00.48 AM.png

From the above Venn diagram, A' = {a, c, f, g, h}.

Question 5. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the sets A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9}, find the complements.

(a) A'

(b) B'

(d) (A ∪ B)'

(e) (A')'

Answer

d. De Morgan's Law

For any sets A and B,

(A ∪ B)' = A' ∩ B'

(A ∩ B)' = A' ∪ B'

Example 5. Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and the sets A = {1, 2, 5} and B = {1, 3, 5, 7, 9}, find the sets using De Morgan's law.

(a) A' ∩ B'

(b) A' ∪ B'

By the De Morgan's law,

(a) A' ∩ B' = (A ∪ B)' = ({1, 2, 3, 5, 7, 9,})' = {4, 6, 8, 10}

(b) A' ∪ B' = (A ∩ B)' = ({1, 5})' = {2, 3, 4, 6, 7, 8, 9, 10}

Question 6. Given the universal set U = {a, b, c, d, e, f, g, h} and the sets A = {a, e, h}, B = {b, c, d, e, h} and C = {b, c, d}, find the complements using De Morgan's law.