Edexcel Mathematics (4XMAH) -Unit 1 - 1.5 Set Language and Notation- Study Notes- New Syllabus
Edexcel Mathematics (4XMAH) -Unit 1 – 1.5 Set Language and Notation- Study Notes- New syllabus
Edexcel Mathematics (4XMAH) -Unit 1 – 1.5 Set Language and Notation- Study Notes -Edexcel iGCSE Mathematics – per latest Syllabus.
Key Concepts:
A understand sets defined in algebraic terms, and understand and use subsets
If A is a subset of B, then A ⊂ B
B use Venn diagrams to represent sets and the number of elements in sets
C use the notation n(A) for the number of elements in the set A
D use sets in practical situations
Edexcel iGCSE Mathematics -Concise Summary Notes- All Topics
Sets and Subsets
A set is a collection of distinct objects, usually numbers.
Sets are written using curly brackets.
\( A=\{1,2,3,4\} \)
Set Builder (Algebraic) Notation
Instead of listing all elements, a set can be defined using a rule.
\( A=\{x\mid x\text{ is an even number less than }10\} \)
This means:
\( A=\{2,4,6,8\} \)
The symbol \( \mid \) means “such that”.
Subsets
A subset is a set whose elements are all contained in another set.
If \( A \) is a subset of \( B \), we write \( A\subset B \).
Example:
\( B=\{1,2,3,4,5,6\} \)
\( A=\{2,4,6\} \)
Every element of \( A \) is inside \( B \), so \( A\subset B \).
Important Symbols
\( \in \) means “is an element of”
\( \notin \) means “is not an element of”
\( \subset \) means “is a subset of”
Key Idea
All elements of a subset must belong to the larger set.
Example 1:
Write the set \( A=\{x\mid x\text{ is a multiple of 3 less than }10\} \) in list form.
▶️ Answer/Explanation
Multiples of 3 below 10 are \( 3,6,9 \)
Conclusion: \( A=\{3,6,9\} \).
Example 2:
Let \( B=\{1,2,3,4,5,6,7,8\} \). Is \( A=\{2,4,6\} \) a subset of \( B \)?
▶️ Answer/Explanation
All elements of \( A \) appear in \( B \).
Conclusion: Yes, \( A\subset B \).
Example 3:
Given \( C=\{x\mid x<5,\; x\in\mathbb{N}\} \), list the elements of \( C \).
▶️ Answer/Explanation
Natural numbers less than 5 are \( 1,2,3,4 \).
Conclusion: \( C=\{1,2,3,4\} \).
Venn Diagrams
A Venn diagram is used to represent sets visually.
Each set is shown as a circle inside a rectangle (the universal set).
The rectangle represents all possible elements.
Universal set: \( U \)
Placing Elements in a Venn Diagram
Numbers belonging to a set go inside its circle.
Numbers outside all circles but inside the rectangle belong to the universal set only.
Intersection
The overlapping region shows elements common to both sets.
\( A\cap B \) means elements in both \( A \) and \( B \).
Union
The union contains everything in either set.
\( A\cup B \)
Number of Elements
We can count how many values are in each region of the diagram.
Each number must be written in the correct region.
Example 1:
Universal set \( U=\{1,2,3,4,5,6,7,8\} \)
\( A=\{2,4,6,8\} \) (even numbers)
\( B=\{3,6\} \) (multiples of 3)
▶️ Answer/Explanation
Common element: \( 6 \) goes in the overlap.
\( 2,4,8 \) go in set \( A \) only.
\( 3 \) goes in set \( B \) only.
\( 1,5,7 \) go outside both circles.
Conclusion: Elements are placed according to membership.
Example 2:
If 5 elements are in \( A \) only, 2 in the intersection, and 4 in \( B \) only, how many are in \( A\cup B \)?
▶️ Answer/Explanation
\( 5+2+4=11 \)
Conclusion: \( n(A\cup B)=11 \).
Example 3:
A Venn diagram shows 3 elements in the overlap, 6 in \( A \) only and 5 in \( B \) only. How many elements are in total inside the circles?
▶️ Answer/Explanation
\( 6+3+5=14 \)
Conclusion: 14 elements.
Number of Elements in a Set
The notation \( n(A) \) means the number of elements in set \( A \).
It tells us how many members the set contains.
Example:
\( A=\{2,4,6,8\} \)
\( n(A)=4 \)
Universal Set
The universal set contains all possible elements and is written \( U \).
\( n(U) \) is the total number of elements in the diagram.
Intersection
Elements common to both sets are written:
\( n(A\cap B) \)
Union
All elements in either set:
\( n(A\cup B) \)
Important Rule
\( n(A\cup B)=n(A)+n(B)-n(A\cap B) \)
We subtract the intersection because it would otherwise be counted twice.
Example 1:
\( A=\{1,3,5,7\} \). Find \( n(A) \).
▶️ Answer/Explanation
There are 4 elements.
Conclusion: \( n(A)=4 \).
Example 2:
If \( n(A)=9 \), \( n(B)=7 \), and \( n(A\cap B)=3 \), find \( n(A\cup B) \).
▶️ Answer/Explanation
\( 9+7-3=13 \)
Conclusion: \( n(A\cup B)=13 \).
Example 3:
The universal set has 20 elements. If 12 are in \( A \) and 5 are outside \( A \), find \( n(A) \).
▶️ Answer/Explanation
Elements outside \( A =5 \)
So elements in \( A =20-5=15 \)
Conclusion: \( n(A)=15 \).
Using Sets in Practical Situations
Sets are often used to organise real-life information such as subjects studied, sports played or products bought.
Venn diagrams help us find:
- how many people are in each group
- how many belong to both groups
- how many belong to neither group
Key Formula
\( n(A\cup B)=n(A)+n(B)-n(A\cap B) \)
This formula prevents counting the same people twice.
Neither Group
To find how many are in neither set:
\( \mathrm{Neither}=n(U)-n(A\cup B) \)
Steps to Solve Problems
1. Identify each set.
2. Fill the intersection first.
3. Fill the remaining parts of each set.
4. Calculate the number outside both sets.
Example 1:
In a class of 30 students, 18 study Maths, 12 study Physics and 5 study both. How many study at least one subject?
▶️ Answer/Explanation
\( n(A\cup B)=18+12-5=25 \)
Conclusion: 25 students study at least one subject.
Example 2:
There are 40 people. 22 like tea, 15 like coffee and 6 like both. How many like neither?
▶️ Answer/Explanation
First find union:
\( 22+15-6=31 \)
Neither \( =40-31=9 \)
Conclusion: 9 people like neither drink.
Example 3:
In a group of 50 students, 28 play football, 24 play cricket and 10 play both. How many play only football?
▶️ Answer/Explanation
Only football \( =28-10=18 \)
Conclusion: 18 students.