Section A — Chapters 1–22

Introduction to Sets

Junior Cycle — 1st Year

✓By the end of this lesson students will be able to define a set and identify its elements.
✓By the end of this lesson students will be able to use correct set notation, including curly braces, the empty set symbol, and the cardinal number notation.
✓By the end of this lesson students will be able to identify subsets and proper subsets.
✓By the end of this lesson students will be able to represent sets and their relationships using Venn diagrams.
✓By the end of this lesson students will be able to find the intersection, union, complement, and difference of sets.

Key concepts

What is a Set?

A set is a well-defined collection of distinct objects. These objects are called the elements or members of the set. Sets are usually denoted by capital letters, and their elements are listed within curly braces { } and separated by commas.

Describing a Set

Sets can be described in two main ways:\n1. **Listing Method (Roster Method)**: Listing all the elements of the set. For example, the set of vowels is V = {a, e, i, o, u}.\n2. **Descriptive Method**: Describing the property that all elements in the set share. For example, P = {prime numbers less than 10}.

Empty Set (Null Set)

The empty set is a set containing no elements. It is denoted by the symbol ∅ or { }.

Cardinal Number of a Set

The cardinal number of a set A, denoted as |A| or #A, is the number of distinct elements in the set A.

Subsets

A set A is a subset of a set B if every element of A is also an element of B. This is denoted by A ⊆ B.\nIf A is a subset of B, and A is not equal to B (meaning B contains at least one element not in A), then A is a proper subset of B. This is denoted by A ⊂ B.

Venn Diagrams

Venn diagrams are visual representations used to show the relationships between sets. A rectangle usually represents the universal set (U), and circles within the rectangle represent individual sets.

Intersection of Sets

The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements that are common to both A and B. In a Venn diagram, this is the overlapping region of the circles for A and B.

Union of Sets

The union of two sets A and B, denoted by A ∪ B, is the set of all elements that are in A or in B (or in both). In a Venn diagram, this includes all elements within both circles.

Universal Set

The universal set, denoted by U, is the set of all elements under consideration in a particular context. It contains all possible elements relevant to the problem.

Complement of a Set

The complement of a set A, denoted by A' (or sometimes Aᶜ), is the set of all elements in the universal set U that are not in A. In a Venn diagram, this is the area inside the universal set rectangle but outside the circle for A.

Set Difference

The difference of set A and set B, denoted by A \ B (or A - B), is the set of all elements that are in A but not in B. It's like taking elements of B away from A.

Key facts to remember

1A set is a collection of distinct objects.
2Elements in a set are listed inside curly braces { }.
3The empty set ∅ or { } contains no elements.
4The cardinal number |A| is the number of elements in set A.
5A ⊆ B means A is a subset of B (all elements of A are in B).
6A ∩ B means the intersection of A and B (elements common to both).
7A ∪ B means the union of A and B (all elements from A or B or both).
8A' means the complement of A (elements in the universal set U but not in A).
9A \ B means the set difference (elements in A but not in B).

Worked examples

Example 1

Let A = {1, 2, 3, 4, 5} and B = {3, 5, 7, 9}.\n(i) Find |A|.\n(ii) List all subsets of {1, 2}.\n(iii) Find A ∩ B.\n(iv) Find A ∪ B.

I(i) The cardinal number of A is the count of elements in A.

II|A| = 5

III(ii) To list all subsets of {1, 2}, we consider sets with 0, 1, or 2 elements from {1, 2}.

IVSubsets: ∅, {1}, {2}, {1, 2}

V(iii) The intersection A ∩ B contains elements common to both A and B.

VIElements common to {1, 2, 3, 4, 5} and {3, 5, 7, 9} are 3 and 5.

VIIA ∩ B = {3, 5}

VIII(iv) The union A ∪ B contains all elements from A and B, without repetition.

9Elements in A: 1, 2, 3, 4, 5

10Elements in B: 3, 5, 7, 9

11Combining them and removing duplicates: 1, 2, 3, 4, 5, 7, 9

12A ∪ B = {1, 2, 3, 4, 5, 7, 9}

Answer

(i) |A| = 5\n(ii) Subsets of {1, 2} are ∅, {1}, {2}, {1, 2}\n(iii) A ∩ B = {3, 5}\n(iv) A ∪ B = {1, 2, 3, 4, 5, 7, 9}

Remember to list elements in a set only once, even if they appear in multiple original sets when finding the union.

Example 2

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set.\nLet P = {prime numbers} and E = {even numbers}.\n(i) List the elements of P and E.\n(ii) Find P'.\n(iii) Find E \ P.

I(i) Identify prime numbers and even numbers within the universal set U.

IIPrime numbers in U: 2, 3, 5, 7

IIIP = {2, 3, 5, 7}

IVEven numbers in U: 2, 4, 6, 8, 10

VE = {2, 4, 6, 8, 10}

VI(ii) The complement of P (P') includes all elements in U that are not in P.

VIIElements in U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

VIIIElements in P: {2, 3, 5, 7}

9Elements in U but not in P: 1, 4, 6, 8, 9, 10

10P' = {1, 4, 6, 8, 9, 10}

11(iii) The set difference E \ P includes all elements that are in E but not in P.

12Elements in E: {2, 4, 6, 8, 10}

13Elements in P: {2, 3, 5, 7}

14Elements in E that are not in P: 4, 6, 8, 10 (Note: 2 is in both, so it's excluded)

15E \ P = {4, 6, 8, 10}

Answer

(i) P = {2, 3, 5, 7}, E = {2, 4, 6, 8, 10}\n(ii) P' = {1, 4, 6, 8, 9, 10}\n(iii) E \ P = {4, 6, 8, 10}

Remember that 1 is not a prime number. The complement always refers to elements within the defined universal set.

Example 3

In a class of 30 students, 18 students play football (F), and 12 students play hurling (H). 5 students play both football and hurling. How many students play neither sport?

IDraw a Venn diagram with two overlapping circles, one for Football (F) and one for Hurling (H), inside a rectangle for the Universal Set (U).

IIThe total number of students (U) = 30.

IIIStart by filling in the intersection: 5 students play both sports.

IVNumber of students who play only Football = (Total Football players) - (Players of both)

VOnly Football = 18 - 5 = 13

VINumber of students who play only Hurling = (Total Hurling players) - (Players of both)

VIIOnly Hurling = 12 - 5 = 7

VIIITotal students playing at least one sport = (Only Football) + (Only Hurling) + (Both)

9Total playing at least one sport = 13 + 7 + 5 = 25

10Number of students who play neither sport = (Total students) - (Total playing at least one sport)

11Neither sport = 30 - 25 = 5

Answer

5 students play neither sport.

Always start filling a Venn diagram from the innermost overlapping region (the intersection) and work outwards. This prevents double-counting.

Common mistakes

✗Forgetting to use curly braces { } when listing elements of a set.
✗Listing elements more than once in a set, especially when finding the union.
✗Confusing the symbols for intersection (∩) and union (∪).
✗Not correctly identifying the universal set when finding the complement of a set.
✗Incorrectly calculating the number of elements in 'only A' or 'only B' in Venn diagram problems by not subtracting the intersection.

Exam tips

★Always read the question carefully to identify the universal set and the specific sets involved.
★Draw Venn diagrams for problems involving two or more sets; they help visualise the relationships and solve problems more easily.
★When solving problems with Venn diagrams, always fill in the intersection (the 'both' region) first.
★Double-check your answers, especially for cardinal numbers and complements, to ensure all elements are accounted for within the universal set.

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