Welcome GCSE Maths

After this week’s lesson, I will be able to What will I learn today? The notation associated with Venn diagrams to evaluate statements written in this notation. After this week’s lesson, I will be able to To recall the meanings of the symbols and notation associated with Venn diagrams. To identify regions in a Venn diagram, using the notation. To be able to assess the truth of statements written in this notation.

SORTING DATA VENN DIAGRAMS

These numbers needed to be sorted into multiples of 3 and even numbers A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Even numbers These numbers needed to be sorted into multiples of 3 and even numbers 3 5 7 10 13 15 18 24 25 27 30 36 40 50

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Even numbers Multiples of 3 go inside this circle 3 5 7 10 13 15 18 24 25 27 30 36 40 50

A VENN DIAGRAM IS USED TO SORT THINGS Even numbers Multiples of 3 Even numbers go inside this circle 3 5 7 10 13 15 18 24 25 27 30 36 40 50

Numbers that are multiples of 3 and even numbers go here... A VENN DIAGRAM IS USED TO SORT THINGS Even numbers Multiples of 3 Numbers that are multiples of 3 and even numbers go here... ...because they belong to both circles. 3 5 7 10 13 15 18 24 25 27 30 36 40 50

A VENN DIAGRAM IS USED TO SORT THINGS Even numbers Multiples of 3 Numbers that are not multiples of 3 and not even, stay on the outside. 3 5 7 10 13 15 18 24 25 27 30 36 40 50

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 5 Multiples of 3 We need to sort these numbers into either multiples of 3 or multiples of 5 3 5 25 30 15 40 17 23 27 33 55 36 18 45 60 12 2 100

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Multiples of 5 Multiples of 3 go inside this circle 3 5 25 30 15 40 17 23 27 33 55 36 18 45 60 12 2 100

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Multiples of 5 Multiples of 5 go inside this circle 3 5 25 30 15 40 17 23 27 33 55 36 18 45 60 12 2 100

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Multiples of 5 NUMBERS THAT ARE MULTIPLES OF 3 AND 5 GO HERE SO THEY ARE IN BOTH CIRCLES 3 5 25 30 15 40 17 23 27 33 55 36 18 45 60 12 2 100

A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Multiples of 5 27 3 25 30 17 40 33 15 5 36 45 55 60 18 2 100 23 12 NUMBERS THAT ARE NOT MULTIPLES OF 3 OR 5 STAY OUTSIDE THE CIRCLES 3 5 25 30 15 40 17 23 27 33 55 36 18 45 60 12 2 100

A VENN DIAGRAM IS USED TO SORT THINGS Which numbers go in each space? Multiples of 3 Multiples of 6 multiples of 10 multiples of 6&10 multiples of 3&10 multiples of 3,6&10 multiples of 3&6 multiples of 6 multiples of 3 Which numbers go in each space? Multiples of 10 3 5 7 10 13 15 18 24 25 27 30 36 40 50

No number here is a multiple of 6, without being a multiple of 3 too! A VENN DIAGRAM IS USED TO SORT THINGS Multiples of 3 Multiples of 6 The numbers remaining don’t belong in any circle, so stay on the outside. Multiples of 10 No number here is a multiple of 6, without being a multiple of 3 too! 3 5 7 10 13 15 18 24 25 27 30 36 40 50

Basic Ideas A set is a collection of objects, which are called the elements or members of the set. A set can be described by listing all the members of the set, or by giving a rule to describe the members. The list or rule is enclosed by braces { }.

Basic Ideas The number of elements in the set A is written as n(A) Sets can be infinite in size, for example the set of prime numbers. P(A) means the probability that an element chosen at random from the universal set, is an element of A. The universal set is everything inside the rectangle. The universal set contains all the elements being discussed in a particular problem, and is shown as a rectangle. The Universal set is indicated by the symbol ɛ and non-membership by the ɛ symbol.

ɛ This Venn diagram shows the number of elements in sets A and B. Find the number of elements in A. Find the probability of picking an element from A. 12 12 19 ɛ A B 7 5 4 3

UNION A union you will know about is the United Kingdom. This is the union of England, Northern Ireland, Scotland & Wales.

Basic Ideas The union of two sets A and B is the set of elements that belong to both sets, and is written A ∪ B Where two or more sets, A and B, overlap is called the intersection, and is written A ∩ B

A ∩ B and A ∪ B A ∩ B means the intersection of A and B – that is, the overlap between A and B or the elements that lie in both A and B. A ∪ B means the union of A and B, which consists of all the elements of A and B put together, including those that are in both sets. The union is where all elements of the sets join together. A ∩ B A ∪ B

Using A’, A ∩ B or A ∪ B, ɛ, n(A) and P(A) This Venn diagram shows the number of elements in sets A and B. Find: P(A ∩ B) n(A ∪ B) ɛ A B 5 7 4 7 19 3 16

' – the complement A' means “the complement of A” – that is, everything except A. An easy way to find this is to cover up A; everything that is left over is A' . A A'

Exam style question

Exam style question

Basic Ideas – Empty Set ∅ means no elements or the empty set. It’s a zero with a line through it. We cannot just write a zero for the empty set. A = ∅ means that A has no elements.

Re-cap The intersection is where two sets overlap. This means in A and in B. A B If you put two sets together, you get the union. This means A or B. A B The complement of A is the region that is not A. This means not A. Definitions:

Objectives Learners will be able to... Define sets of numbers by describing or by listing. Explain the terms ‘universal’, ‘null or empty’, ‘Intersection’ and ‘Union’. Use correct notation when working with sets.