Sets and Venn Diagrams Explained for Students

Sets are one of the first topics in Matric and O Level maths, and they come back in probability, statistics and university discrete maths. Venn diagrams turn set problems into pictures, which makes even tricky survey questions easy to organise. This guide walks you through the notation, the operations and the classic word problems.

What Is a Set?

A set is a well-defined collection of distinct objects, called elements. "Well-defined" means you can always decide whether something belongs to the set or not.

• \(A = \{2, 4, 6, 8\}\) is a set.
• "The tall students in my class" is not a set, because "tall" is not clearly defined.

We write \(4 \in A\) (4 belongs to \(A\)) and \(5 \notin A\).

Ways to Describe a Set

1. Tabular (roster) form: list the elements, e.g. \(\{1, 3, 5, 7\}\).
2. Set-builder form: describe a rule, e.g. \(\{x \mid x \text{ is an odd number}, x < 8\}\).
3. Descriptive form: in words, "the set of odd numbers less than 8."

Important Kinds of Sets

• Empty set \(\varnothing\) or \(\{\}\): has no elements.
• Finite and infinite sets.
• Subset: \(A \subseteq B\) if every element of \(A\) is also in \(B\).
• Universal set \(U\): contains all elements under discussion.
• Power set \(P(A)\): the set of all subsets of \(A\). If \(A\) has \(n\) elements, it has \(2^n\) subsets.

For example, \(A = \{a, b, c\}\) has \(2^3 = 8\) subsets, including \(\varnothing\) and \(A\) itself.

Set Operations

• Union \(A \cup B\): elements in \(A\) or \(B\) (or both).
• Intersection \(A \cap B\): elements in both \(A\) and \(B\).
• Difference \(A - B\): elements in \(A\) but not in \(B\).
• Complement \(A'\): elements of \(U\) that are not in \(A\).

Two sets are disjoint if \(A \cap B = \varnothing\).

De Morgan's Laws

$$ (A \cup B)' = A' \cap B' \qquad (A \cap B)' = A' \cup B' $$

Venn Diagrams

A Venn diagram shows the universal set as a rectangle and each set as a circle inside it. Overlapping regions represent intersections. Shading a region shows the result of an operation. For two sets there are four regions: only \(A\), only \(B\), both, and neither.

Worked Example 1: Set Operations

Let \(U = \{1, 2, 3, \ldots, 10\}\), \(A = \{2, 4, 6, 8, 10\}\) and \(B = \{1, 2, 3, 4, 5\}\). Find \(A \cup B\), \(A \cap B\), \(A - B\), \(A'\), and verify \((A \cup B)' = A' \cap B'\).

Step 1: Union. Combine all elements without repeating:

$$ A \cup B = \{1, 2, 3, 4, 5, 6, 8, 10\} $$

Step 2: Intersection. Common elements: \(A \cap B = \{2, 4\}\).

Step 3: Difference. Remove elements of \(B\) from \(A\): \(A - B = \{6, 8, 10\}\).

Step 4: Complements.

$$ A' = \{1, 3, 5, 7, 9\}, \qquad B' = \{6, 7, 8, 9, 10\} $$

Step 5: Verify De Morgan's law.

• Left side: \((A \cup B)' = U - \{1, 2, 3, 4, 5, 6, 8, 10\} = \{7, 9\}\).
• Right side: \(A' \cap B' = \{1, 3, 5, 7, 9\} \cap \{6, 7, 8, 9, 10\} = \{7, 9\}\).

Both sides equal \(\{7, 9\}\), so the law is verified.

The Counting Formula

For two sets:

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

We subtract the intersection because it was counted twice. For three sets:

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

Worked Example 2: Two-Set Survey Problem

In a class of 60 students, 35 study Maths, 28 study Physics and 10 study neither. How many study both? How many study only Maths?

Step 1: Students studying at least one subject:

$$ n(M \cup P) = 60 - 10 = 50 $$

Step 2: Use the formula:

$$ 50 = 35 + 28 - n(M \cap P) \quad \Rightarrow \quad n(M \cap P) = 63 - 50 = 13 $$

Step 3: Only Maths \(= 35 - 13 = 22\). Only Physics \(= 28 - 13 = 15\).

Check: \(22 + 13 + 15 + 10 = 60\). Correct.

Worked Example 3: Three-Set Problem

In a survey of 100 students, 45 play football (F), 50 play cricket (C) and 30 play hockey (H). Also, 20 play F and C, 12 play C and H, 10 play F and H, and 5 play all three. How many play none? How many play exactly two games?

Step 1: Total playing at least one game:

$$ n(F \cup C \cup H) = 45 + 50 + 30 - 20 - 12 - 10 + 5 = 88 $$

Step 2: None \(= 100 - 88 = 12\).

Step 3: Fill the Venn diagram from the centre outwards. Exactly two games:

• F and C only: \(20 - 5 = 15\)
• C and H only: \(12 - 5 = 7\)
• F and H only: \(10 - 5 = 5\)

Exactly two \(= 15 + 7 + 5 = 27\).

Step 4: Check with "only" regions. Only F \(= 45 - 15 - 5 - 5 = 20\), only C \(= 50 - 15 - 7 - 5 = 23\), only H \(= 30 - 7 - 5 - 5 = 13\). Then \(20 + 23 + 13 + 27 + 5 = 88\). Correct.

Tips

• Always start filling a Venn diagram from the innermost region (the intersection of all sets) and work outwards.
• Read carefully: "play F and C" usually includes those who play all three; "only F and C" does not.
• Always check that all regions add up to the total.
• Remember that \(\varnothing\) is a subset of every set.

Key Takeaways

• A set is a well-defined collection; a set with \(n\) elements has \(2^n\) subsets.
• Union means "or", intersection means "and", complement means "not".
• \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\), with an extended version for three sets.
• De Morgan's laws link complements of unions and intersections.
• In Venn diagram problems, fill from the centre outwards and verify the total.

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