Set Language and Set Notation | O Level Mathematics 4024 & IGCSE Mathematics 0580 | Detailed Free Notes To Score An A Star (A*)

Lesson Objectives

Sets

• When you collect data together
  • And can identify it.
  • It is called as set.
• Important terms
  • Each individual part of the set is called an element
  • For example, if there are the numbers 2, 5, 8 is a set.
    • Then 2 is an element of the set
    • 5 is an element of the set
    • 8 is an element of the set
  • Universal set
    • The universal set is the complete set or data within which all other sets are.
      • In other words, each set is a subset of the universal set.
      • All the elements of the subset are contained in the universal set
      • Why is this important
        • For example, the examiner tells you the Universal set, denoted by E, has first 100 prime numbers.
        • Now, in this question, ALL you subjects can ONLY prime numbers
          • Why?
            • Because the subsets can not have elements that are not contained in the universal set.
            • So if the next line says, set A has first 10 prime numbers, this set is possible in the given situation as universal set has first 100 prime numbers.
            • But if it says, in this context, that set B has first 10 positive numbers, this is not possible because not all the first 10 positive numbers are prime numbers.
        • Usually
          • In O Level, the universal set has EVERYTHING in it. So there are no restrictions USUALLY for the subset.
• Do these definitions come in exams
  • No, but knowing them helps a lot when solving questions
• How can the elements be mentioned
  • There are two ways
    • First, you can use the set notation to list the elements.
      • How?
        • The set always starts ad end with these brackets {}
        • In between, you list the elements and separate each element using a comma
      • For example
        • The set containing the first 5 positive integers will be as follows
        • {1,2,3,4,5}
    • You can explain the set
      • This is the way the examiner will give you the set
      • For example
        • The set A contains the first five prime numbers
• Always remember
  • Set is denoted by capital letter
  • So it will be Set A not Set a
  • Also, you usually put a = sign between set and its elements
  • For example
    • If Set A has first 5 positive integers
    • You will write
      • A = {1,2,3,4,5}

Common Set Notations

• You must know the meaning and functions of some common set notations
• n (A)
  • Whenever you write n before a set, basically you are telling the total number of elements in the set.
  • For example
    • If a set is A= {1,2,3}
    • Then n{A} = 3 because there are a total of 3 elements in the set.
• ∈
  • This sign means “element of”
  • Please make sure you read if from left to right
  • Thus
    • 10 ∈ A
    • means that 10 is an element of the set A
    • NOT THE OTHER WAY AROUND
    • So A is not an element of set 10
• ∉
  • Not an element of
  • As with the previous one, read from left to right
  • So
    • 10 ∉ A
    • Means that 10 is not an element of A
• ⊂
  • Subset of
  • Again read from left to right
  • Some what tricky one
    • Basically, it means that the set on the left of the sign ⊂ is a subject of the set on the right of the sign
    • Meanings
      • All the elements of set on the left are already contained in the elements in the set on the right
    • For example
      • A = {1,2,3,4,5}
      • B = {1,2,3}
    • Here, B ⊂ A because all elements of B are already in set A
• ⊃
  • Basically, it is the reverse of the one given above
  • Super set of
  • Read from left to right
  • A ⊃ B
  • Means that B is a subset of A / A is the super set of B
    • Same concept, all elements of B are already contained in A
• ⊇
  • This one means subset of or EQUAL TO
  • Remember, this one again is read from LEFT TO RIGHT.
  • If it says
    • B ⊇ A
    • It means that not only A has all the elements of B
    • But B and A are the same
    • S
    • A = {1,2,3} and B = {1,2,3}
  • Why is this important?
    • If the examiner tells you A is a subset of B but A is not a subset equal to B, you will know that B has some elements other than A ones as well

Types of Sets

• Finite Set
  • A set is finite if you can exactly determine the number of elements in it
  • In other words, the number of elements are not infinite.
  • For example
    • A = {1,2,3,4,5}
    • Here, A is a finite set as we know there are 5 elements in A
• Infinite Set
  • A set where the elements are infinite
  • You can not determine the exact number of elements
  • For example
    • A = {x: x is a prime number}
  • Now, you can determine the total number here, as prime numbers are infinite.

The Important Relations

• The most tricky part of this chapter
• Extremely crucial part for Venn Diagrams
  • Complement Set
    • It is the set that contains ALL THE VALUES not contained in the original set.
    • Also called prime set (HAS NOTHING TO DO WITH PRIME NUMBERS – just a name)
    • We denote it using ‘ sign
    • So
    • If set A has all the positive integers
    • set A’ (or A complement or A prime set) will have all the non positive integers or negative integers
    • Similarly if set A has all prime numbers
    • set A’ will have the opposite ALL NON PRIME NUMBERS
  • Equal Set
    • The name suggests it
    • Being completely equal
  • Null set or empty set
    • SUPER IMPORTANT COMMON MISTAKE
      • A = {0} is NOT A NULL or EMPTY SET
        • It is a set with the element 0 in it
    • A null set or empty set on the other hand is shown as follows
      • A = {}
      • Or another method is to write
      • A = Ø
  • Intersection
    • Intersection, in easiest words, mean common elements
    • ∩ this is the sign
    • So if A = {1,2,3,4,5} and B = {1,2,3}
    • Then A ∩ B is {1,2,3} the common elements
    • A tricky question
      • What will be A ∩ B’
      • First we need to find B’
      • B prime is all the numbers except 1,2,3
      • So A and B prime will have {4,5} in common
    • Disjoint intersection
      • In such cases, a common does not exist between two sets
      • For example set A has all even numbers
      • Set B has all odd numbers
      • Then A ∩ B = Ø
      • As they have no common
  • Union
    • In simple words, union means putting two sets together.
    • ∪ is the sign
    • For example A={1,2,3,4,5} and B= {10,11,12}
    • Then A ∪ B is {1,2,3,4,5,10,11,12}