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Spring 2016 COMP 2300

**Discrete Structures for Computation **

**Counting Elements of Disjoint Sets: The Addition Rule**

Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1

• Suppose a finite set

*A*

equals the union of

*k A*

1

*A*

2 ,

*A k*

distinct

*N*

(

*A*

)

*N*

(

*A*

1 )

*N*

(

*A*

2 )

*N*

(

*A k*

).

*A*

1

*A*

3

*A*

2

*A*

4

**Spring 2016 COMP 2300 Donghyun (David) Kim **

2

**Department of Mathematics and Physics North Carolina Central University**

**Counting Password with Three or Fewer Letters**

• A computer access password consists of from one to three letters chosen from the 26 in the alphabet with repetitions allowed. How many different passwords are possible?

*A*

1 number of passwords of length 1

*A*

2 number of passwords of length 2

*A*

2

*A*

1

*A*

3

*A*

3 number of passwords of length 3

**Spring 2016 COMP 2300 Donghyun (David) Kim **

3

**Department of Mathematics and Physics North Carolina Central University**

**Counting Password with Three or Fewer Letters – cont’**

• A computer access password consists of from one to three letters chosen from the 26 in the alphabet with repetitions allowed. How many different passwords are possible?

number of passwords of length 1 26 number number the of of passwords passwords number of of length 2 26 2 of length 3 26 3 passwords 26 26 2 26 3

**Spring 2016 COMP 2300 Donghyun (David) Kim **

4

**Department of Mathematics and Physics North Carolina Central University**

**Counting the Number of Integers Divisible by 5**

• How many three-digit integers are divisible by 5?

*A*

1 three that end in 0

*A*

1

*A*

2

*A*

2 three that end in 5

**Spring 2016 COMP 2300 Donghyun (David) Kim **

5

**Department of Mathematics and Physics North Carolina Central University**

**Counting the Number of Integers Divisible by 5 – cont’**

• How many three-digit integers are divisible by 5?

9 choices (1 ~ 9) 10 choices (0 ~ 9) 2 choices (0 or 5)

*N*

(

*A*

1 )

*N*

(

*A*

2 ) 9 10 9 10 9 10 2 180

**Spring 2016 COMP 2300 Donghyun (David) Kim **

6

**Department of Mathematics and Physics North Carolina Central University**

• If

*A N*

(

*A*

is a finite set and

*B*

*B*

)

*N*

(

*A*

)

*N*

is a subset of (

*B*

).

*A*

, then

*A*

(

*n*

elements)

*B*

(

*k*

elements)

*A*

*B*

(

*n*

*k*

elements)

**Spring 2016 COMP 2300 Donghyun (David) Kim **

7

**Department of Mathematics and Physics North Carolina Central University**

**Counting PINs with Repeated Symbols**

• The PINs are made from exactly four symbols chosen from the 26 letters of the alphabet and the ten digits, with repetitions allowed.

• How many PINs contain repeated symbols?

• If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?

**Spring 2016 COMP 2300 Donghyun (David) Kim **

8

**Department of Mathematics and Physics North Carolina Central University**

**Counting PINs with Repeated Symbols – cont’**

• The PINs are made from exactly four symbols chosen from the 26 letters of the alphabet and the ten digits, with repetitions allowed.

• How many PINs contain repeated symbols?

• • • Total possible cases: # of cases without any repetition: Solution: 36 36 36 36 36 36 36 36 36 35 36 35 34 34 33 33 • If all PINs are equally likely, what is the probability that a randomly chosen PIN contains a repeated symbol?

36 36 36 36 36 36 36 36 35 36 34 33 9

**Spring 2016 COMP 2300 Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University**

**Formula for the Probability of the Complement of an Event**

• If

*S P*

( is a finite sample space and

*A A C*

) 1

*P*

(

*A*

).

is an event in

*S*

, then

*P*

(

*A*

)

**Spring 2016 COMP 2300 Donghyun (David) Kim **

*P C*

(

*A*

) 10

**Department of Mathematics and Physics North Carolina Central University**

• Theorem 9.3.3: The Inclusion/Exclusion Rule for Two or Three Sets • If

*A*

,

*B*

, and

*C N*

(

*A*

*B*

)

*N*

are any finite sets, then (

*A*

)

*N*

(

*B*

)

*N*

(

*A*

*B*

) and

*N*

(

*A*

*B*

*C*

)

*N*

(

*A*

)

*N*

(

*B*

)

*N*

(

*C*

)

*N*

(

*A*

*B*

)

*N*

(

*B*

*C*

)

*N*

(

*A*

*C*

)

*N*

(

*A*

*B*

*C*

)

**Spring 2016 COMP 2300 Donghyun (David) Kim **

11

**Department of Mathematics and Physics North Carolina Central University**

**Counting Elements of a General Union**

• How many integers from 1 through 1,000 are multiple of 3 or multiples of 5?

• How many integers from 1 through 1,000 are neither multiples of 3 nor multiples of 5?

• 1000 – 467 = 533 12

**Spring 2016 COMP 2300 Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University**

**Counting Elements of a General Union – cont’**

• How many integers from 1 through 1,000 are multiple of 3 or multiples of 5?

• # of multiple of 3: 3, 6, …, 999 = 3 (1, 2, … , 333) • # of multiple of 5: 5, 10, …, 1000 = 5 (1, 2, … , 200) • # of multiple of 15: 15, 30, …, 990 = 15 (1, 2, …, 66) • Answer: 200 + 333 – 66 = 533 = 467 • How many integers from 1 through 1,000 are neither multiples of 3 nor multiples of 5?

• 1000 – 467 = 533 13

**Spring 2016 COMP 2300 Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University**

• •

**Counting the Number of Elements in an Intersection**

Out of a total of 50 students in the class, • 30 took precalculus • 18 took calculus • 26 took java • 9 took both precalculus and calculus • 16 took both precalculus and java • 8 took both calculus and java • 47 took at least one of the three courses How many students did not take any of the three courses?

• How many students took all three courses?

• How many students took precalculus and calculus but not java?

**Spring 2016 COMP 2300 Donghyun (David) Kim **

14

**Department of Mathematics and Physics North Carolina Central University**

• • • •

**Counting the Number of Elements in an Intersection**

Out of a total of 50 students in the class, • 30 took precalculus • 18 took calculus • 26 took java • 9 took both precalculus and calculus • 16 took both precalculus and java • 8 took both calculus and java • 47 took at least one of the three courses How many students did not take any of the three courses?

50 47 3 How many students took all three courses?

47 ( 30 18 26 ) ( 9 16 8 ) 6 How many students took precalculus and calculus but not java?

9 6 3 15