Objectives of the Lesson

Dear Learners,

At the end of the lesson, you will be able to:-

• State the Basic Set Operations
• Explore Relationships Between Sets.
• Apply Venn Diagrams to Visualize Operations.
• Solve Real-World Problems

Brain storming questions

Suppose Collect different flower colors in each pair and build sets from those flowers according to their color. Thus, Feyisa collects $\{ red, Tulip, purple, yellow and white\}$ flowers and Semira also collects $\{ red, green, rose, yellow\; and\; white, sunflower, bluebell, and lotus\}$ flowers from flower supermarket. Based on the above sets, build the following sets:

(a) Set of colours of flowers collected by Feyisa or Semira or both of them (Feyisa Union Semira),
(b) Set of flowers of common colours collected by Feyisa and Semira (Feyisa Intersection Semira),
(c) Set of colours of flowers collected by Feyisa only (Feyisa difference/less Semira),

(d) Set of colours of flowers except the colours of flowers collected by Semira (Semira compliment).
(e) Set of colours of flowers collected by Semira but not collected by Feyisa or Set of colours of flowers collected by Feyisa but not collected by Semira (Feyisa symmetric difference Semira)

Key Terms

• Union of sets.
• Intersection of sets.
• Relative Complement.
• Absolute Compliment.
• Cartesian Product.

Union of sets

A union of A and B contains all the elements in A or B or both sets. The set notation used to represent the union of sets is ∪. The set operation, set combination or union, is represented by:

$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$.

Here, x is found in both sets A and B.

Example 1:

Let set $A= \{a, b, c, d\}$ and $B = \{ c, d, e, f, g\}$. then determine $A \cup B$.

Solution

$A \cup B = \{ a, b, c, d, e, f, g \}$

Intersection of sets

The intersection of two sets 𝐴 and 𝐵, is denoted by 𝐴 ∩ 𝐵, is the set of all elements stated in both set 𝐴 and set 𝐵.

We write this in mathematics:

$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

Example 2:

Let set $A= \{ a, b, c, d \}$ and $B = \{ c, d, e, f, g\}$. then determine $A \cap B$.

Solution:

$A \cap B= \{ c, d \}$.

Complement of sets.

Relative Complement (set difference)

The difference of two sets A and B is defined as the lists of all the elements that are in set A but that are not present in set B. The set notation used to represent the difference between the two sets A and B is A − B or A ∖ B.

Example 3:

Let set $A= \{ a, b, c, d \}$ and $B = \{ c, d, e, f, g \}$. then determine i. $A \setminus B$ $\;$ ii. $B \setminus A$.

Solution:

i. $A \setminus B = \{ a, b\}$ $\;$ii. $B \setminus A = \{ e, f, g\}$.

Absolute Compliment ( ‘ )

A’ is the set which contains all the elements of the universal set other than the set itself (set A).

Example 4:

Let set $A= \{ a, b, c, d\}$ and $B = \{ c, d, e, f, g\}$ and $U = \{ a, b, c, d, e, f, g, h\}$, then determine i. $A’$ ii. $B’$

Solution:

i. $A ‘ = \{ e, f, g, h\}$ ii. $B’ = \{ a, b, h\}$.

De Morgan’s Law

For the complement set of 𝐴 ⋃ 𝐵 and 𝐴 ⋂ 𝐵, we have

a. $(A \cup B)’ = (A’ \cap B’)′$,

b. $(A \cap B)’ = (A’ \cup B’)$.

For any two sets 𝐴 and 𝐵, each of the following holds true.
i. $(A’)’ = A$

ii. $A’ = U \ A$

iii. $𝐴 \ 𝐵 = A \cap B’$

iV. $A \subseteq B \implies B’ \subseteq A’$

Symmetric Difference of Two Sets

The set which contains the elements which are either in set A or in set B but not in both is called the symmetric difference between two given sets. It is represented by $A \triangle B$ and is read as a symmetric difference of set A and B.

$A \triangle B = (A \setminus B) \cup (B \setminus A)$
or
$A \triangle B = (A \cup B) \setminus (A \cap B)$

Example 5:

Let set $A = \{ a, b, c, d \}$ and $B = \{ c, d, e, f, g \}$ and $U = \{ a, b, c, d, e, f, g, h\}$ then determine $A \triangle B$,

Solution:

$A \triangle B = \{ a, b, e, f, g\}$.

Venn – Diagram

Venn diagrams are diagrams used to represent sets, diagramming the relationships between sets and operations. Introduced by John Venn (1834-1883), the Venn diagram uses circles (overlapping, intersecting, and disjoint) to show the relationships between circles.

Example 6:

From the Venn diagram above find i. $A\cup B$ ii. $A\cap B$ iii. A/B iv. B/A v. $A\bigtriangleup B$ Vi. A’

Solution:

i. $A \cup B=\{ 1, 2, 3, 4, 5, 7, 9\}$

ii. $A \cap B = \{ 1, 3, 5 \}$

iii. $A \setminus B = \{ 7, 9\}$

iv. $B \setminus A = \{ 2, 4 \}$

V. $A \triangle B = (A \setminus B) \cup (B \setminus A) = \{ 2, 4, 7, 9 \}$.

vi. $A’=\{0,2,4,6,8.10\}$

Example 7:

Construct a Venn – diagram using the sets $A=\{ 0, 1, 4, 5\}$ and $B=\{ 1, 4\}$ and $U=\{0, 1, 2, 3, 4, 5, 6, 7, 8\}$.

Solution:

Cartesian Product of Two Sets

The Cartesian product $S \times T$ of S and T consists of all ordered pairs (s, t) such that s ∈ S and t ∈ T . Ordered pairs are characterized by the following property: $(a, b)=(c, d)$ if and only if. $a=c$ and $b=d$.

Note that: $S \times T$ is not the same as $T \times S$ except $S = T$.

$A=\{a, b, c, d\}$ ,

$B=\{c, d, e, f, g\}$ and

$U=\{a, b, c, d, e, f, g, h\}$. Then we have

$A \times B=\{(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)\}$

$B \times A = \{(1, x), (2, x), (3, x), (1, y), (2, y), (3, y)\}$

In other words, for the ordered pairs $(a, b)$ and $(c, d)$ to be equal, both the first components must be equal (a=c) and the second components must be equal $(b = d)$. This is written as:

$(a, b) = (c,d) \implies   a = c$ and $b = d$.

Example 8:

Find the values of $x \;\text {and} \;y$ if $(2x – 3, 5) = (9, y +2)$

Solution:

$(2x – 3, 5) = (9, y +2)$

$\implies 2x – 3=9$ and $5=y+2$

$2x=12 \; and\; y=5-2$.

Thus, $x = 6 \; and \; y=3$.