HW 1-1 (Due Aug. 25, 2016) Name:

Problem 1. Suppose that A and B are two events. Write expressions involving unions, intersections,

and complements that describe the following:

1. Both events occur

2. At least one occurs

3. Neither occurs

4. Exactly one occurs

1

ANB

AUBAnd

#( An 5) U ( Bna )

Problem 2. suppose a family contains two children of di↵erent ages, and we are interested in the

gender of these children. Let F denote that a child is female and M that the child is male and let

a pair such as FM denote that the older child is female and the younger is male. THere are four

points in the set S of possible observations:

S = {FF, FM,MF,MM}.

Let A denote the subset of possibilities containing no males; B, the subset containing two males;

and C, the subset containing at least one male. List the elements of A, B, C, A [B, A \B, A [C,

A \ C, B [ C, B \ C, and C \ B̄.

2

A={ FF } B={ MM }

C= ( FM ,MF

,MM }

AUB= { FFMM }

AnB=¢Avc=S Ancaf

BVEC BAC=B

cnD= { FM ,MF }

Problem 3. Define the sequence of sets Aj = (1� 1/j, 2 + 1/j), for j = 1, 2, . . . . Then what are

[1j=1Aj and \1

j=1 Aj?

3

¥,Aj = ¥

,Ctjt ,

ztjt ) = ( 0,3 )

f),

Aj = [ 1,2 ]

HW 1-2 (Due Aug. 25, 2016) Name:

Problem 1. If A and B are two sets, draw Venn diagrams to verify the followings:

1. A = (A \B) [ (A \ ¯B)

2. if B ⇢ A then A = B [ (A \ ¯B)

Use the identities A = A \ S and S = B [ ¯B and a distributive law to prove that (mathematically,

not graphically)

1. A = (A \B) [ (A \ ¯B)

2. if B ⇢ A then A = B [ (A \ ¯B)

3. Further, show that (A \ B) and (A \ ¯B) are mutually exclusive and therefore that A is the

union of two mutually exclusive sets, (A \B) and (A \ ¯B).

4. Also show that B and (A \ ¯B) are mutually exclusive and if B ⇢ A, A is the union of two

mutually exclusive sets, B and (A \ ¯B).

1

"

K¥979 .ee#anit

l . CAABIUCANB )=AACBvBT=AnS=A

2 If BCA AAB=B

So from 1:

A=(ArB)UCAdB )

= BUCANB )

3. (AMDALAAD )=CAnA)n( BAD )=An¢=¢

4.

BACAAB )= AACBRB )=An¢=§

Problem 2. Suppose two dice are tossed and the numbers on the upper faces are observed. Let S

denote the set of all possible pairs that can be observed. For example, let (2, 3) denote that a 2 was

observed on the first die and a 3 on the second.

1. Define the following subsets of S: A: The number on the second die is even. B: The sume of

the two numbers is even. C: At least one number in the pair is odd. Using equally likely rule

to calculate P (A), P (B) and P (C).

2. List the points in A,

¯C, A \B, A \ ¯B,

¯A [B and

¯A \ C.

2

SI { ( ij ) : ii. 2.345.6, jil ,

2. 3. 4. 5.6 }

l' PC A) =L

.

PC B) = 3×36+1×4=21

PC c) = 1- PC both are even )=tY¥=¥

2. A={ ( i ,

2k ) : i. I. 2. 3. 4. 5.6,

k=l ,2,3 }

-

f- { ( i,

i ) : iii. 2.3.4516 # uB={ ( i. zjtl ) : 'tt :b

C 3 ,i ) : i=l ,

2. 3. 4. 5,6foil .2

,

C2i ,2jJ . i. I. 2.3

( 5,

i ) :it ,

2. 3. 4.516, II. 2.3 )

( 2 ,i ) : is 1. 3.5

( 4. i ) :i=l ,

3.5 At Atf @ zjtpiiti :b

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AAB = { cziizj ) : E- 1.2 ,} ,jH .

2.3)

An5= { C Ziti ,zj ) :ital '3j⇒2 , } }

Problem 3. Suppose two balanced coins are tossed and the upper faces are observed.

1. List the sample points for this experiment

2. Assign a reasonable probability to each sample point. (Are the sample points equally likely?)

3. Let A denote the event that exactly one head is observed and B the event that at least one

head is observed. List the sample points in both A and B.

4. From your answer to part 3., find P (A), P (B), P (A \B), P (A [B) and P (

¯A [B).

3

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Be { CHHI,

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