Let's Learn Mathematics

Sets Theory

Definition of Sets

A set is a collection of well-defined, distinct objects called elements or members. Sets are typically denoted by capital letters with elements enclosed in curly braces.

Given set A containing odd numbers less than 10:

\[ A = \{1, 3, 5, 7, 9\} \]

Membership Notation:

For any element x:

  • If \( x \) is in set \( A \): \( x \in A \)
  • If \( x \) is not in set \( A \): \( x \notin A \)

Well-Defined:

A set must have unambiguous membership criteria. For example, "books in this room" is well-defined, while "interesting books" is not.

Complement of a Set (\( A' \))

The complement of set \( A \) contains all elements not in \( A \) within a given universal set.

If \( \mathbb{U} = \{1, 2, 3, \ldots, 10\} \) and \( A = \{1, 3, 5, 7, 9\} \), then:

\[ A' = \{2, 4, 6, 8, 10\} \]

Universal Set (\( \mathbb{U} \))

The set containing all elements under consideration. It satisfies:

\[ A \cup A' = \mathbb{U} \]

Cardinality (\( n(A) \))

The number of elements in a set:

\[ n(A) = 5 \] (for our example set)

Finite Set:

Has countable elements (e.g., \( \{1, 2, 3\} \))

Infinite Set:

Has uncountable elements (e.g., \( \mathbb{N} = \{1, 2, 3, \ldots\} \))

\[ n(\mathbb{N}) = \infty \]

Describing Sets

1. Word Description

Using natural language to define the set:

\( B =\) {whole numbers from 20 to 30}

2. Listing (Roster)

Explicitly listing all elements:

\[ B = \{20, 21, 22, \ldots, 30\} \]

3. Set-Builder Notation

Mathematical rule definition:

\[ B = \{x \mid 20 \leq x \leq 30, x \in \mathbb{W}\} \]

Read as: "x such that x is between 20 and 30 inclusive, where x is a whole number"

Types of Sets

Equal Sets

Two sets are equal if they contain exactly the same elements, regardless of order.

Example:

\[ A = \{1, 2, 3\}, \quad B = \{3, 2, 1\} \] \[ A = B \]

Subsets

Set \( A \) is a subset of \( B \) if all elements of \( A \) are in \( B \):

\[ A \subseteq B \]

Example:

\[ A = \{1, 2\}, \quad B = \{1, 2, 3, 4\} \] \[ A \subseteq B \]

Proper Subset:

When \( A \subseteq B \) but \( A \neq B \):

\[ A \subset B \]

Disjoint Sets

Sets with no common elements:

\[ A \cap B = \emptyset \]

Example:

\[ A = \{1, 3, 5\}, \quad B = \{2, 4, 6\} \] \[ A \cap B = \emptyset \]

Special Sets

Empty Set (\( \emptyset \))

Contains no elements:

\[ \emptyset = \{\} \] \[ n(\emptyset) = 0 \]

Unit Set

Contains exactly one element:

\[ S = \{a\} \]

Number Systems

Standard sets of numbers used in mathematics:

Natural Numbers (\( \mathbb{N} \))

\[ \mathbb{N} = \{1, 2, 3, \ldots\} \]

Whole Numbers (\( \mathbb{W} \))

\[ \mathbb{W} = \{0, 1, 2, 3, \ldots\} \]

Integers (\( \mathbb{Z} \))

\[ \mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} \]

Rational Numbers (\( \mathbb{Q} \))

\[ \mathbb{Q} = \left\{\frac{p}{q} \mid p,q \in \mathbb{Z}, q \neq 0\right\} \]

Real Numbers (\( \mathbb{R} \))

All numbers on the number line

Relationships between number systems:

\[ \mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \]

Set Operations

Union (\( A \cup B \))

Combines all elements from both sets:

\[ A \cup B = \{x \mid x \in A \text{ or } x \in B\} \]

Example:

\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \cup B = \{1, 2, 3, 4, 5\} \]

Intersection (\( A \cap B \))

Contains only elements common to both sets:

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

Example:

\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \cap B = \{3\} \]

Difference (\( A \setminus B \))

Elements in \( A \) but not in \( B \):

\[ A \setminus B = \{x \mid x \in A \text{ and } x \notin B\} \]

Example:

\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \setminus B = \{1, 2\} \]

Symmetric Difference (\( A \Delta B \))

Elements in either set but not in both:

\[ A \Delta B = (A \setminus B) \cup (B \setminus A) \]

Example:

\[ A = \{1, 2, 3\}, \quad B = \{3, 4, 5\} \] \[ A \Delta B = \{1, 2, 4, 5\} \]

Venn Diagrams

Visual representation of set relationships:

The Venn diagram

Union:

All shaded areas

Intersection:

Overlapping area

Complement:

Area outside the set

Three-Set Problems

Problems involving three sets with overlapping regions:

The diagram

Cardinality formula for three sets:

$ n(A \cup B \cup C) $ $=$ $n(A)$ $+$ $n(B)$ $+$ $n(C)$ $-$ $n(A \cap B)$ $-$ $n(A \cap C)$ $-$ $n(B \cap C)$ $+$ $n(A \cap B \cap C)$

Practice Exercise

Question 1

Given \( \mathbb{U} = \{1, 2, \ldots, 10\} \) and \( A = \{2, 4, 6, 8, 10\} \), find \( A' \).

\[ A' = \{1, 3, 5, 7, 9\} \]

Question 2

If \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), find \( A \Delta B \).

\[ A \Delta B = \{1, 2, 4, 5\} \]

Question 3

For sets \( X = \{a, b, c\} \) and \( Y = \{c, d, e\} \), verify that $ n(X \cup Y)$ $=$ $n(X)$ $+$ $n(Y)$ $-$ $n(X \cap Y)$.

\(X \cup Y = \{a, b, c, d, e\} \)
\( \Rightarrow n(X \cup Y) = 5 \)
\( n(X) + n(Y) - n(X \cap Y) \)
\( \Rightarrow 3 + 3 - 1 = 5 \)

Question 4

In a class of 50 students:

  • 30 study Math
  • 25 study Physics
  • 20 study Chemistry
  • 10 study Math and Physics
  • 8 study Math and Chemistry
  • 7 study Physics and Chemistry
  • 5 study all three subjects

How many students study none of these subjects?

Question 5

SSCE 1995 Q9b

There are 40 pupils in a class. 30 of them study Biology, 22 study Physics and 21 study Chemistry. 15 study Physics and Biology, 10 study Physics and Chemistry, 13 study Biology and Chemistry. 2 study Physics only, 3 study Chemistry only and 7 study Biology only.

($i$) Represent this information on a Venn diagram.
($ii$) How many pupils study all the three subjects?
($iii$) If a pupil is selected at random, what is the probability that he studies either Physics or Chemistry?

Ready for more challenges?

Full Practice Set


Quick Test

This section contains 100 multiple choice questions. You have 60 minutes to complete it.

Each question has four options labeled A to D. Select the correct answer for each question.

60:00
Question 1 of 100