Mathacle. A; if u is not an element of A, then A. Some of the commonly used sets and notations are

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1 Mathale 1. Definitions of Sets set is a olletion of objets. Eah objet in a set is an element of that set. The apital letters are usually used to denote the sets, and the lower ase letters are used to denote the elements. Given a set, if u is an element of, then u ; if u is not an element of, then u. Some of the ommonly used sets and notations are : Natural numbers (inluding zero, sometimes it is alled whole numbers) : Integers : Positive integers : Rational numbers : Real numbers : Complex numbers : Empty set Example 1.1. Some examples: Notations Sometimes the sets an be desribed by listing the elements is olletion of a, b,, d is {0,1, 2, 3,...} T { a, b,, d} The rule based set-builder notations are used to desribe the sets that are not so easily expressed by listing. For instane, the set S onsists of all real numbers between 1and 1, inlusive, is S { x x, 1 x 1} The empty set is the set with no elements. It is denoted by or {}. 1

2 Example 1.2. Mathale S { x x, x 0 and x 0} Set and set B are eual if they have exatly the same elements. It is denoted as B. If set and set B do not have exatly the same elements, then set and set B are not eual, or B. Example 1.2. If T { a, b,, d} and R { d, a,, b}, then T R. If a set S is finite, then the number of elements in the set S is denoted as ns ( ). The above example, nt ( ) 4. Subsets and Universal Set Set B is the subset of set if for every element in B, it is also in set. This is denoted as B. For any partiular problem, the universal set U is the set that ontains all the subsets involved with the problem. If is a subset of U, the set of elements of U that are not in is alled the omponent of, denoted by. Example 1.3. If set { a, b,, d, e}, B { b, }, U { a, b,, d, e, f, g}, then B, { f, g}, B { a, d, e, f, g}. Example 1.4. If set C { a, b,, d}, then the set S ontains all the subsets of C are S {, { a},{ b},{ },{ d }, { a, b},{ a, },{ a, d},{ b, },{ b, d},{, d }, { a, b, },{ a, b, d},{ a,, d},{ b,, d }, { a, b,, d }} If the number of element is ns ( ) for set S, then there are ns ( ) 4, and there are For any set, and. subsets. ( ) 2 ns subsets. In this example, 2. Operations on the Events If and B are two events of the universal set U, then 1.) does not our is the omplement of. It is denoted by ; 2

3 Mathale 2.) either or B our is the union of and B. It is denoted by B ; 3.) both and B our is the intersetion of and B. It is denoted by B ; 4.) ours and B does not our is the differene of and B. It is denoted by B B. Example 2.1. In the experiment of rolling a die one, the universal set is U {1, 2,3,4,5,6}. Let = an odd number turns up B = the number that turns up is divisible by 3 Express the following events in terms of and B: E = the number that turns up is even? F = the number that turns up is odd or is divisible by 3? G = the number that turns up is odd and divisible by 3? H = the number that turns up is odd but not divisible by 3? 3

4 Mathale Solution: {1,3, 5}, B {3, 6}, and E is the omplement of : E U {2,4,6} F is the union of and B: F B {1,3, 5,6} G is the intersetion of and B: G B {3} H is the differene of and B: H B B {1,5} Notie that E U and E { }. In general, two events and B are exhaustive if B U, in partiular U. That is, either or B will our. Two events and B are disjoint or mutually exlusive if B, where is the impossible event, in partiular,. That is, if ours, then B an not our. Example 2.2. Toss a oin three times. Let H= Heads and T= Tails. Express the following sets in terms of H and T: The universal set U = we throw tails exatly two times B = we throw tails at least two times C = tails did not appear before a heads appeared D = two heads E : F ( C D) : G D : 4

5 Mathale Solution: U { HHH, HHT, HTH, THH, HTT, TTH, THT, TTT} { HTT, THT, TTH} B { HTT, THT, TTH, TTT} C { HTT, HTH, HHT} D { HHT, THH, HTH} E { HHH, HHT, HTH, THH, TTT} F { HTT, THT, TTH, HHT, HTH} G { HTT, THT, TTH} Example 2.3. If it is hot and B it is sunny, state the following sets in terms of and B: 1.) B 2.) B 3.) B 4.) B 5.) B 6.) B B 5

6 Mathale Solution: 1.) B it is hot and sunny 2.) B it is not hot but sunny 3.) B it is neither hot nor sunny 4.) B it is not true that it is neither hot nor sunny 5.) B it is hot or sunny B B it is either not hot and sunny or hot and not sunny 6.) Example 2.4. p It is raining Mary is sik t Bob stayed up late last night r Paris is the apital of Frane s John is a loud-mouth Express eah of the following statements in terms of the delarative sentenes above: Negation: 1.) It isn t raining 2.) It is not the ase that Mary isn t sik 3.) Paris is not a apital of Frane 4.) John is in no way a loud-mouth Conjuntion: 5.) It is raining and Mary is sik 6.) Bob stayed up late last night and John is a loud-mouth 7.) Paris isn t the apital of Frane and It isn t raining 8.) John is a loud-mouth but Mary isn t sik 6

7 Mathale 9.) It is not the ase that it is raining and Mary is sik Disjuntion: 10.) It is raining or Mary is sik 11.) Paris is the apital of Frane and it is raining or John is a loud-mouth 12.) Mary is sik or Mary isn t sik 13.) John is a loud-mouth or Mary is sik or it is raining 14.) It is not the ase that Mary is sik or Bob stayed up late last night Mixed statements: 15.) It is raining but Mary is not sik 16.) Either it is not raining and Mary is sik or Paris is not the apital of Frane 17.) Neither it is raining nor Mary is sik 18.) It is not true that both it is raining and Mary is sik Solution: 1.) p 2.) ( ) 3.) r 4.) s 5.) p 6.) t s 7.) r p 8.) s 9.) p 10.) p 11.) 12.) r p s 7

8 Mathale 13.) ( s ) p 14.) ( t) 15.) p 16.) 17.) p r p 18.) p 3. DeMorgan s Laws and Properties of Set Operations DeMorgan s Laws. For any two events and B, Example 3.1. B B and B B Let J = John is to blame and M = Mary is to blame. Use DeMorgan s Laws to show event and event B are euivalent, where = It is ertainly not true that neither John nor Mary is to blame B = John or Mary is to blame, or both Hints: find Solution: J and M first and express and B in terms of J, M, J and M. J M John is not to blame Mary is not to blame J M J M B 8

9 Mathale Properties and Laws of Sets Commutative: B B ssoiative: ( B C) ( B) C Distributive: ( B C) ( B) ( C) ( B C) ( B) ( C) Partition: B B exlusive., where B and B are mutually Other properties: ( ) B B B ( B ) B Example 3.2. Prove eah identity a.) B B ( B ) B ( B ) B ( B ) B ( B ) B ( B ) ( B B ) ( B ) B B 9

10 Mathale b.) ( B ) B ( B ) ( B ) ( B) ( ) B.) ( B) C ( B C) ( B) C ( B ) C ( B C ) ( B C) ( B C) d.) ( B ( B )) B ( B ( B )) B ( B ) B ( B ) B e.) ( B) B ( B) ( B) ( B ) ( B) ( ) ( B) B 10

11 Mathale f.) ( B) ( B ) ( B) ( B) ( B) ( B ) ( B ) ( B ) (( B ) B) ( B ) ) (( B) ( B B)) (( ) ( B )) (( B) ) ( ( B )) ( B) ( B ) ( B) ( B ) ( B) ( B ) ( B) ( B) 11

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2 ( : Focus on free Education) Exercise 16.2 Question 1: A die is rolled. Let E be the event die shows 4 and F be the event die shows even number. Are E and F mutually exclusive? Answer 1: When a die is