Set Notation & Venn Diagrams | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Set Notation

What is a set? 

• A set is a group of elements. These elements can vary and might include numbers, letters, coordinates, and more.
• You can represent a set by listing its elements within braces {}. For example, {1, 2, 3, 6} represents a set containing the factors of 6.
• When elements in a set follow a specific rule, you can express this using a colon within the braces {... : ...}. The section before the colon signifies the type of element, while the part after the colon denotes the rule.
• Sets commonly consist of numbers, but they can also include coordinates. For instance, {(x, y): y = 2x - 1} describes a set of points lying on the line y = 2x - 1.

What do I need to know about set notation?

• ℰ is the universal set (the set of everything)
  • e.g.  if talking about factors of 24 then ℰ = {1, 2, 3, 4, 6, 8, 12, 24}
  • You may see alternative notations used for calligraphic E 
    • 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}
• 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 the complement of A
  • This is thought of as “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 visual tool used to show relationships between sets and their elements.
• It consists of a rectangle that represents the universal set (ℰ) and circles that represent individual sets.
  • Circles in a Venn diagram can overlap to show common elements between sets.

What do the different regions mean on a Venn diagram? 

• A' is represented by the regions that are not in the A circle
• A ∩ B is represented by the region where the A and B circles overlap
• A ∪ B is represented by the regions that are in A or B or both