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Set Notation & Venn Diagrams (Edexcel IGCSE Maths)
Revision Note
Set Notation
What is a set?
- A set is a collection of elements
- Elements could be anything - numbers, letters, coordinates etc
- You could describe a set by writing its elements inside curly brackets {}
- e.g. {1, 2, 3, 6} is the set of the factors of 6
- If the set of elements follow a rule then you can write this using a colon inside the curly brackets {... : ...}
- The bit before the colon is just the type of element
- The bit after the colon is the rule
- e.g. {x : x2 < 100} is the set of numbers which are less than 100 when squared
- The elements are usually numbers but these could be coordinates
- e.g. {(x, y) : y = 2x + 1} is the set of points that lie on the line y = 2x + 1
What do I need to know about set notation?
- is the universal set (the set of everything)
- e.g. if talking about factors of 24 then = {1, 2, 3, 4, 6, 8, 12, 24}
- You may see alternative notations used for
- U is a common alternative
- S or the Greek letter ξ (xi) may also be seen
- ∅ is the empty set (the set of with no elements}
-
- e.g. {x : x is an even prime bigger than 5} = ∅ as there are no even primes bigger than 5
- We normally use upper case letters to represent sets (A, B, C, ...) and lower case letters to represent elements (a, b, c, ...)
- n(A) is the number of elements in set A
- e.g. n({1, 4, 9}) = 3
- Note n(∅) = 0 as there are no elements in the set but n({0}) = 1 as there is 1 element in the set
- a ∈ A means a is an element of A (a is in the set A)
- e.g. If x ∈ {1, 4, 9} then x is either 1, 4 or 9
- A ⊆ B means A is a subset of B
- This means every element in A is also in B
- e.g. {students in class Y that pass the exam} ⊆ {students in class Y}
- A ⊂ B means A is a proper subset of B
- This means A is a subset of B but not the same as B (A ⊆ B but A ≠ B)
- The difference between ⊆ and ⊂ is similar to the difference between ≤ and <
- e.g. {1, 2, 3, 6} ⊂ {1, 2, 3, 4, 6, 8, 12, 24}
- Putting a cross through the symbol means it is not true
- Similar to ≠ meaning not equal
- a ∉ A means a is not an element of the set A
- A ⊈ B means A is not a subset of B
- A ⊄ B means A is not a proper subset of B
- A ∩ B means the intersection of A and B (the overlap of A and B)
- This is the set of elements that are in both set A and set B
- A ∪ B means the union of A and B (everything in A or B or both)
- This is the set of elements that are in at least one of sets
- A' is “not A” (everything outside A)
- This is the set of elements that are not in A
Sets & Venn Diagrams
What is a Venn diagram?
- A Venn diagram is a way to illustrate all the elements within sets and any intersections
- A Venn diagram consists of
- a rectangle representing the universal set ()
- a circle for each set
- Circles may or may not overlap depending on which elements are shared between sets
What do the different regions mean on a Venn diagram?
- is represented by the regions that are not in the A circle
- is represented by the region where the A and B circles overlap
- is represented by the regions that are in A or B or both
Worked example
Two sets A and B are shown in the Venn diagram.
The elements of A are anything that is inside the A circle. A = {2, 6, 12, 14, 28}. There are 5 elements in it.
n(A) = 5
14 and 28 are the elements that are in both A and B.
{14, 28} = A∩B
A' is the set of elements not in A so A' = {1, 5, 7, 8, 21, 35}.
B = {7, 14, 21, 28, 35}.
A'∪B is the set of elements that are in at least one of the sets.
A'∪B = {1, 5, 7, 8, 14, 21, 28, 35}
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