3.1 Set Notation

1. U U B X A 3.1 Set Notation Example 1 X  U (b) A  B 3.1.1 Venn Diagrams Venn Diagram is used to illustrate the idea of sets and subsets.

2. Y X • (c)there exists an element a such that a  X and a  Y. or (X  Y and Y  X)

3. 3.1.2 Operations on Sets • Intersection • Union • Complement

4. Intersection X  Y = {x : x  X and x  Y} where  means “intersection”. Example 1 Given X = {2,3,4,6,7,8, 10} and Y= {4,5,-2, 6, 9, 10}. Find X  Y.

5. Y X X and Y are disjoint if X Y = . • where  denotes the empty set. • When X and Y are disjoint, the Venn Diagram of X  Y is

6. Y X U Union X  Y = {x : x  X or x  Y} • The Venn Diagram is

7. Example 1 Let A = {3,5,8,9,10} and E = {12,4, 3, 5, 10,24, 9}. Find A E and AE.

8. Complement • We use or to denote the complement of X. • In addition, we use Y\X to denote the relative complement of X w.r.t. Y. = {x : x  U and x  X} Y\X = {y : y  Y and y  Y}

9. U A B U B A • Example 1 • Please mark in the following diagrams to indicate the relative complement of A w.r.t. B.

10. Example 1 • Consider a deck of playing cards. Let U be the set of all the cards. R be the set of all the red cards. D be the set of all the diamond cards. What , D\R and R\D? Illustrate these sets with a Venn diagram.

11. 3.2 Number of Elements • For any two sets A and B , we have: n(AB) = n(A) + n(B) – n(AB)

12. Example 1 • Of the 70 S6 students of a school, 39 studied Mathematics and Statistics(M), 37 studied Geography (G), 42 studied History (H), 24 studied both M and G, 26 studied both M and H, 25 studied both G and H, 18 studied all three subjects. • Find the number of students who study (a) at least one of the three subjects, (b) none of the three subjects.

13. 3.3 Probability • Relative Frequency Definition of Probability Suppose that a random experiment is repeated a large number of times N, and that the event A occurs n times. Then the probability of A is the limiting value of the relative frequency as N becomes very large. • Weaknesses of the relative frequency definition • It requires a large number of repetitions of an experiment to establish the probability of an even. • It assumes that the relative frequency will tend to a LIMIT.

14. Some Properties of Probability For every event A in the sample sapce S, • 0  P(A)  1 • P(S) = 1 • If A and B are mutually exclusive events in S, then P(AB) = P(A) + P(B) *** P(impossible event) = 0 P(the certain event) = 1

15. Law for Complementary Events P(A’) = 1 – P(A) Example • A card is drawn at random from an ordinary pack of 52 playing cards. Find the probability that the card (a) is a seven, (b) is not a seven.

16. 3.4 Methods of Counting • The Multiplication Principle • [Please refer to your F.6 Textbook] • Permutations • [Please refer to your F.6 Textbook] • Combinations • [Please refer to your F.6 Textbook

17. Combinations • Example 1 • If the letters of the word “MINIMUM” are arranged in a line at random, what is the probability that the three M’s are together at the beginning of the arrangement? • Example 2 • Ten pupils are placed at random in a line. What is the probability that the two youngest pupils are separated? • Example 3 • If a four-digit number is formed form the digits 1,2,3 and 5 and repetitions are NOT allowed, find the probability that the number is divisible by 5? • Example 4 • In how many ways can a hand of 4 cards be dealt from an ordinary pack of 52 palying cards?

18. Example 5 • Four letters are chose at random from the word RANDOMLY. Find the probability that all four letters chosen are consonants. • Example 6 • A team of 4 is chosen at random from 5 girls and 6 boys. • In how many ways can the team be chosen if (i) there is are no restrictions; (ii) there must be more boys than girls? • Find the probability that the team contains only one boy. • Example 7 • Four items are taken at random from a box of 12 items and inspected. The box is rejected if more than 1item is found to be faulty. If there are 3 faulty items in the box, find the probability that the box is accepted.