Sets SCIE Centre Additional Maths © Adam Gibson

Aims: To understand the idea of a set To be able to use the appropriate mathematical symbols (such as ) to describe sets To be able to use Venn diagrams and make calculations

A set is any collection of distinct objects. Give me FOUR members of each of these sets: B = All natural numbers which are a multiple of 2 but not a multiple of 4 Can you tell me SETS ?

SET NOTATION Say these aloud: A union B A intersection B A is a proper subset of B A is a subset of B x is not a member of A The number of elements of A The complement of A The null set The universal set x is a member of A

DESCRIBING SETS These are the ELEMENTS or MEMBERS of C

Some Basic Definitions Definition of a Set We define a set as a collection of objects with the property that, given an arbitrary object, it is possible to tell whether or not that object belongs to the set. Definition - Equality of Sets Two sets A and B are said to be equal, written A = B, if they have the same elements. Definitions – Subset If A and B are sets, B is said to be a subset of A if every element of B is also an element of A. That is, B ⊆ A if x ∈ B ⇒ x ∈ A

Historical Aside Bertrand Russell tried to formalise Mathematics based on logic. However, he came across a problem… Is the set of all sets which are not members of themselves a member of itself? “Russell’s paradox” 1+1=2

CONCEPT CHECK … What is ? A: The null set has NO elements, so the answer is zero. What is ? A: The number of elements in the universal set will depend on the problem (often it will be infinite). True or false? A: True. The null set is a proper subset of any other set, by definition.

In the box is every student in the school. M for kids in your MATHS class S for kids in your SCIENCE class G for kids in your GYM class

Students in Math OR ScienceStudents in Math AND Science Students in Math AND Science AND Gym Students NOT in Gym or Math

Students NOT in Gym AND Math In other words: you start to tell a joke in math class, but the bell rings And you have to finish it in gym. Who DOESN’T get the joke? These guys heard the whole joke Sooo everyone else

AB C DE F

ABCDEFABCDEF Answers

IS SET THEORY USEFUL? Question Of the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone.  40 of them had both a two-wheeler and a credit card.  30 had both a credit card and a mobile phone.  60 had both a two wheeler and mobile phone.  10 had all three. How many candidates had none of the three?

NOTEWORTHY RESULTS Hence, solve the problem and draw a Venn diagram

SOLUTION T = two wheelers M = mobile phones C = credit cards

SOLUTION Question Of the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone.  40 of them had both a two-wheeler and a credit card.  30 had both a credit card and a mobile phone.  60 had both a two wheeler and mobile phone.  10 had all three. So there are 10 job applicants with none of the three.

PRACTICE TASKS 1 Convert these English statements to set notation a)The number of elements in C AND D is 24 b)The number of elements in either A or B is 5. c)Set X is the intersection of sets Y and Z d)Sets A and B have no common elements e)The number of elements in neither A nor B is 1 f)G is not a proper subset of H 2) a) List all subsets of A = {5,6,9} b) How many subsets does B have, B = {x:x<20, x is prime} 3) Let k, x and y all be natural numbers. We define S(k) as the set of number pairs (x,y) as follows: Plot a graph of n(S(k)) against k. Can you find the equation for n(S(k)) as a function of k? 4) Cantor says that the number of elements in the set {1,2,3,4…} is the same as the number of elements in the set {2,4,6,8…}. Is he right?