Sets

Core Concepts

Description of Sets A set is a well-defined collection of distinct objects. The objects contained in a set are called its elements or members. We typically denote sets with capital letters (e.g., $A, B, C$) and elements with lowercase letters (e.g., $a, b, c$).

• Membership: If an element $x$ belongs to a set $A$, we write $x \in A$. If it does not belong, we write $x \notin A$.
• Representation: Sets are usually represented by listing elements inside curly braces, e.g., $A = \{2, 4, 6, 8\}$, or by stating a defining rule (set-builder notation), e.g., $A = \{x \mid x \text{ is an even number between 1 and 9}\}$.
• Universal Set ($\xi$ or $U$): The master set containing all possible elements under consideration in a particular problem.

Types of Sets

• Finite Set: A set with a countable number of elements (e.g., $\{1, 2, 3\}$). The number of elements in set $A$ is denoted as $n(A)$.
• Infinite Set: A set with an uncountable, endless number of elements (e.g., the set of all integers).
• Empty (Null) Set: A set containing no elements, denoted by $\emptyset$ or $\{\}$.
• Singleton Set: A set containing exactly one element (e.g., $\{5\}$).
• Disjoint Sets: Two sets that have no elements in common.

Subsets If every element of set $A$ is also an element of set $B$, then $A$ is a subset of $B$, denoted as $A \subseteq B$.

• Proper Subset ($A \subset B$): $A$ is a subset of $B$, but $A \neq B$ (i.e., $B$ has at least one element not in $A$).
• Number of Subsets: If a set has $n$ elements, the total number of subsets it can form is given by the formula $2^n$. The empty set and the set itself are always subsets of the set.

Operations with Sets

• Union ($A \cup B$): A set containing all elements that are in $A$, in $B$, or in both.
• Intersection ($A \cap B$): A set containing only the elements that are common to both $A$ and $B$.
• Difference ($A - B$): A set containing elements that are in $A$ but strictly not in $B$. Note that $A - B$ is equivalent to $A \cap B'$.
• Complement ($A'$): A set containing all elements in the universal set $\xi$ that are not in $A$.

Venn Diagrams A Venn diagram is a visual representation of sets. The universal set $\xi$ is drawn as a rectangle, and subsets are drawn as circles within it. For any two sets $A$ and $B$ within a universal set:

• The region where circles overlap represents $A \cap B$.
• The total area covered by both circles represents $A \cup B$.
• The area outside both circles represents $(A \cup B)'$, containing elements in neither $A$ nor $B$.

A fundamental formula relating the number of elements in two intersecting sets is: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ When factoring in the universal set: $$n(\xi) = n(A \cup B) + n((A \cup B)')$$

Worked Examples

Example 1: Basic Set Operations and Venn Diagrams Given the universal set $\xi = \{15, 30, 45, 60, 75\}$, $A = \{15, 45\}$ and $B = \{30, 60\}$. (a) Find $(A \cup B)'$. (b) Draw a Venn diagram to represent the sets.

Solution: (a) First, find the union of $A$ and $B$, which combines all elements from both sets: $$A \cup B = \{15, 30, 45, 60\}$$ Next, find the complement of the union, $(A \cup B)'$. These are elements in $\xi$ that are not in $A \cup B$: $$(A \cup B)' = \{75\}$$

(b) To draw the Venn diagram:

1. Draw a rectangle for $\xi$.
2. Draw two disjoint circles for $A$ and $B$ (since they share no common elements, $A \cap B = \emptyset$).
3. Place $15$ and $45$ inside circle $A$.
4. Place $30$ and $60$ inside circle $B$.
5. Place $75$ inside the rectangle but outside both circles.

Example 2: Venn Diagram Word Problem with Algebra In a certain school, $40$ students were asked whether they like tennis or football or both. It was found that the number of students who like both tennis and football was three times the number of students who like tennis only. Furthermore, the number of students who like football only was $6$ more than twice the number of students who like tennis only. However, $4$ students like neither tennis nor football. Represent this information in a Venn diagram, letting $x$ be the number of students who like tennis only, and find the value of $x$.

Solution: Let $\xi$ be the set of all students asked, $T$ be the set of students who like tennis, and $F$ be the set of students who like football. Given:

• $n(\xi) = 40$
• Number of students who like tennis only = $x$
• Number of students who like both ($T \cap F$) = $3x$
• Number of students who like football only = $2x + 6$
• Number of students who like neither $(T \cup F)' = 4$

Venn Diagram Setup: Draw a rectangle with two intersecting circles $T$ and $F$.

• Region $T$ only: $x$
• Intersection $T \cap F$: $3x$
• Region $F$ only: $2x + 6$
• Outside circles: $4$

To find $x$, sum all distinct regions and equate them to the universal set total: $$x + 3x + (2x + 6) + 4 = 40$$ $$6x + 10 = 40$$ $$6x = 30$$ $$x = 5$$ The number of students who like tennis only is $5$.

Example 3: Solving Without a Venn Diagram In a village of 50 farmers, 25 grow cashew nut, and 16 grow both cashew nut and maize. If 10 farmers grow neither cashew nut nor maize, find the number of farmers who grow maize only. Do not use a Venn diagram.

Solution: Let $\xi$ be the set of all farmers, $C$ be the farmers who grow cashew nuts, and $M$ be the farmers who grow maize. Given:

• $n(\xi) = 50$
• $n(C) = 25$
• $n(C \cap M) = 16$
• $n((C \cup M)') = 10$

We know that the total number of farmers is the sum of those who grow at least one crop and those who grow neither: $$n(\xi) = n(C \cup M) + n((C \cup M)')$$ $$50 = n(C \cup M) + 10$$ $$n(C \cup M) = 40$$

Using the set operation formula for two sets: $$n(C \cup M) = n(C) + n(M) - n(C \cap M)$$ Substitute the known values: $$40 = 25 + n(M) - 16$$ $$40 = 9 + n(M)$$ $$n(M) = 31$$ This is the total number of farmers who grow maize. However, we need the number of farmers who grow maize only. $$\text{Maize only} = n(M) - n(C \cap M)$$ $$\text{Maize only} = 31 - 16 = 15$$ Therefore, $15$ farmers grow maize only.

NECTA Exam Focus

When tested on the CSEE Basic Mathematics exam, the Sets topic follows highly predictable patterns:

• Word Problems with Two Sets: The vast majority of questions require modeling real-world situations (e.g., sports, subjects, crops) using two intersecting sets. Expect to be explicitly instructed to either use or not use a Venn diagram. If a method is specified, failing to follow the instruction will result in lost marks.
• Algebraic Venn Diagrams: NECTA frequently introduces unknowns. A common pattern is describing regions in terms of another (e.g., "three times the number of students who like tennis only"). Students must confidently assign a variable like $x$ to a region, express other regions algebraically, and solve the linear equation by summing all distinct regions to equal $n(\xi)$.
• The "Neither" Region: A very common pitfall is forgetting to include the items outside the intersecting circles (the complement of the union). Always ensure that the sum of 'A only', 'B only', 'Both', and 'Neither' equals the total universal set.
• Careful Interpretation of Language: Distinguishing between "$A$" and "$A \text{ only}$" is critical. If a question states "30 study Chemistry", this represents $n(C) = 30$, which includes those who study Chemistry and another subject. If it says "30 study Chemistry only", this strictly refers to the region outside the intersection.
• Set Definitions and Subsets: You will occasionally encounter shorter questions requiring you to calculate subsets (using $2^n$), list elements bounded by a condition, or find the percentage of a set's members out of a universal set.

Practice Problems

1. In a school of $60$ teachers, some drink Fanta and some drink Coca-Cola. If $46$ drink Fanta, $18$ drink Coca-Cola and $14$ drink both Coca-Cola and Fanta. How many teachers drink neither Fanta nor Coca-Cola? (Use a Venn diagram).
2. In a class of $45$ students, $30$ study Chemistry, $20$ study Physics and $5$ study neither of the two subjects. Represent this information in a well-labeled Venn diagram.
3. Use the following information about a universal set $U$ and its disjoint sets $A$ and $B$: $U = \{a, b, c, d, e, f, g, h\}$, $A = \{c, f, g\}$, and $B = \{d, b, h\}$. Find the number of subsets of set $B'$.
4. Find the percentage of numbers which are multiples of 5 from the set $\{1, 2, 3, 4, \ldots, 52\}$. Write the answer correct to one decimal place.

Subtopics

• Description of sets
• Types of sets
• Subsets
• Operations with sets
• Venn diagrams

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.