1.2: Sets

Sets:

1. What is a Set?

• A set is a collection of distinct objects or numbers. We usually write sets using curly brackets { }.

• Notation: A = {1, 2, 3} and the objects in a set are called elements or members.

• Example: a set of even numbers less than 10: {2, 4, 6, 8}

Practice Problem

Write the set of odd numbers less than 10.

Answers are in the end

2. Element of a Set

• Shows whether something belongs to a set.

• Symbol: ∈ meaning: “is an element of”

• Example: 3 ∈ A (3 is in set A).

• Symbol: ∉ meaning: “is not an element of”

• 5 ∉ A (5 is not in set A).

• Example: if B = {a, b, c}, then b ∈ B.

Practice Problem

Is 7 ∈ {2, 4, 6, 8}?

3. Number of Elements in a Set

• Represents how many elements are in the set.

• Notation: n(A) meaning: Number of elements in set A.

• Example: if A = {2, 4, 6, 8}, then n(A) = 4.

Practice Problem

Find n(C) if C = {1, 3, 5, 7, 9}.

4. Empty Set

• A set with no elements at all.

• Notation: ∅ or {} meaning: the set contains nothing.

• Example: the set of prime numbers between 8 and 10 = ∅ (because 9 is not prime).

Practice Problem

What is the set of whole numbers less than 0?

5. Universal Set

• The set of all possible elements in a particular context.

• Notation: Usually shown by the symbol ξ (capital Greek letter Xi) or described in words.

• Example: if studying numbers from 1 to 10, then universal set ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

Practice Problem

Write a universal set for all even numbers ≤ 10.

6. Subset

• A set where all elements are contained within another set.

• Notation: A ⊆ B meaning: A is a subset of B.

• Example: 

  • A = {2, 4}

  • B = {2, 4, 6, 8}

  • A ⊆ B.

Practice Problem

Is {3, 5} a subset of {1, 3, 5, 7}?

7. Complement of a Set

• All elements in the universal set that are not in A.

• Notation: A′ meaning: complement of A.

• Example: 

  • Universal set ξ = {1, 2, 3, 4, 5}

  • A = {1, 3, 5}

  • A′ = {2, 4}

Practice Problem

If ξ = {1, 2, 3, 4} and B = {2, 4}, find B′.

8. Union of Sets

• Combines all elements from two sets without repeating.

• Notation: A ∪ B meaning: A union B.

• Example:

  • A = {1, 2, 4}

  • B = {2, 3, 5}

  • A ∪ B = {1, 2, 3, 4, 5}

Practice Problem

Find A ∪ B if A = {a, b} and B = {b, c}.

9. Intersection of Sets

• Elements common to both sets.

• Notation: A ∩ B meaning: A intersection B.

• Example:

  • A = {1, 2, 3}

  • B = {2, 3, 4}

  • A ∩ B = {2, 3}

Practice Problem

Find A ∩ B if A = {x, y, z} and B = {y, z, w}.

10. Venn Diagrams

• A visual way to show relationships between sets.

• Features:

  • Circles represent sets.

  • Overlaps show intersection (∩).

  • The whole rectangle = universal set.

• Example:

• A circle for A with {2, 4}.

• A circle for B with {4, 6}.

• Overlap shows {4}.

Practice Problem

Draw a Venn diagram showing:

A = {2, 4}

B = {4, 6}

 A = {x : x is a natural number}

• The curly braces { } mean “the set of all things inside”.

• The colon : is read as “such that”.

• “A = {x : x is a natural number}” is read as: set A is the set of all x such that x is a natural number.

• In words: set A contains all natural numbers.

• Reminder: natural numbers (ℕ): 1, 2, 3, 4, 5, ...

• Full Example of A: A = {1, 2, 3, 4, 5, 6, …}

Practice Problem

Write the first five elements of set A.

B = {(x, y) : y = mx + c}

• The set B contains pairs written as (x, y).

• The rule y = mx + c defines the relationship between x and y.

• In words: B is the set of all coordinate pairs (x, y) that satisfy the equation of a straight line y = mx + c.

• Explanation:

  • m is the slope (gradient).

  • c is the y-intercept.

  • Any (x, y) pair on the line follows y = mx + c.

• Example: if m = 2 and c = 3, then:

  • y = 2x + 3

  • B = {(x, y): y = 2x + 3}

• Example Elements in B:

  • (0, 3) ∈ B (because 2×0+3=3)

  • (1, 5) ∈ B (2×1+3=5)

Practice Problem

If y = 3x + 1, write 2 pairs (x, y) that belong to B.

C = {x : a ⩽ x ⩽ b}

• Set C contains all values of x.

• The rule a ⩽ x ⩽ b means x is between a and b inclusive.

• In words: C is the set of all numbers x that are greater than or equal to a and less than or equal to b.

• Explanation: includes a and b themselves and is all numbers in the interval [a, b].

• Example:

  • If a = 2 and b = 5 so C = {x : 2 ⩽ x ⩽ 5}

  • x can be 2, 3, 4, 5 (if integers) so C = {2, 3, 4, 5}

  • Or any number between 2 and 5 (if real numbers): 2.1, 3.5, etc. so C = all real numbers from 2 to 5 inclusive.

Practice Problem

If a = 0 and b = 3, write all integer elements of set C.

Answers to Practice Problems

1. Odd numbers < 10: {1, 3, 5, 7, 9}

2. Is 7 ∈ {2, 4, 6, 8}? No (7 ∉ the set)

3. n(C) = 5

4. Set of whole numbers < 0 = ∅

5. Universal set for even numbers ≤ 10: {2, 4, 6, 8, 10}

6. {3, 5} ⊆ {1, 3, 5, 7}: Yes

7. B′ = {1, 3}

8. A ∪ B = {a, b, c}

9. A ∩ B = {y, z}

10. 2 circles drawn. In circle A: you'll have 2, in circle B you'll have 6, and in the intersection you'll have 4

11. First five elements of A: 1, 2, 3, 4, 5

12. For y = 3x + 1: 

• • (0, 1) (since 3×0+1=1)

  • (2, 7) (3×2+1=7)

  • If a = 0, b = 3: 0, 1, 2, 3

13. C= {0, 1, 2, 3}

NOTES DONE BY FARIDA SABET

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