DISCRETE STRUCTURE

Bs (IT)

DISCRETE STRUCTURE

Question no 1What are truth value of those that are proposition ?a) Boston is a capital of Massachusetts.

Ans.) proposition . T

b) Miami is capital of Florida.Ans.) proposition . T

c) 2+3=5 ans.) proposition . T

d) 5+7=10 ans.) proposition .F

e) x+2=11 no proposition

SLIDE # 1

f) Answer this question .

Ans.) no propositionQ2: Which of these are proposition ? What are

the truth value of these that are proposition?a: Do not pass go.

Ans.) no proposition b: What time is it .

Ans.) no propositionc: There are no black flies in Maine.

Ans.) proposition. There are black flies in Maine

d: 4+x=5Ans.) no proposition

e: The moon is made of green cheese.Ans.) proposition

f: 2n ≥ 100Ans.) no proposition

Q3:What is negation of each of these proposition?

Slide # 2

SLIDE # 3

a) Today is Thursday. Ans.) Today is not Thursday

b) There is no pollution in new jersey Ans.) There is pollution in new jersey

c) 2+1=3Ans.) 2+1 is not equal to 3

d) The summer in Maine is hot and sunnyAns.) The summer in Maine is hot and sunny

SLIDE # 4

Q4: Let p and q be the proposition ?p: I bought a lottery ticket this week .q : I won the million dollar jackpot on Friday.Express each of these proposition as an English

sentences?1:~ pAns.) I not bought a lottery ticket this week

2: p ν qAns.) I bought a lottery ticket this week or I won the

million dollar jackpot on Friday

SLIDE # 5

c) P → qAns.) If I bought a lottery this week then I won million dollar

jackpot on Friday

d) p Λ qAns.) I bought a lottery ticket this week and I won million

dollar jackpot on Friday

e)p ↔ q Ans.) I bought a lottery ticket this week if and only if I won

million dollar jackpot on Friday

f) ~p ↔ q

SLIDE # 6

Ans.) if I not bought a lottery ticket this week then I not win million dollar jackpot on Friday

g) ~ p Λ ~qAns.) I not bought a lottery ticket this week

and I not win million dollar jackpot on Friday

Q5: Let p and q be proposition “swimming at new jersey shore is allowed and sharks have been spotted near the shore” respectively

Express each of these compound proposition in an English sentences

SLIDE # 7

a) ~ pAns.) Swimming at new jersey shore is not allowed

b) p Λ qAns.) swimming at new jersey shore is allowed and

sharks have been spotted near shore

c) p → ~ qAns.) if swimming at new jersey shore is allowed the

sharks have not been spotted near shore

d) P ↔ ~qAns.) swimming at new jersey shore is allowed if and

only if the sharks have not been spotted near shore

SLIDE # 8

Q6: Let p and q be proposition “ the election is decided and the votes have been counted respectively .Express each of these compound proposition as an English sentences

a) q → pAns.) if the votes have been counted then the

election is decided.

b) p ↔ qAns.) the election is decided if and only if votes have been counted

SLIDE # 9

Q7:Let p and q proposition p: it is below freezing .q :it is snowing . Write these proposition using p and q and logical

connectives.a) It is below freezing and snowing .Ans.) p Λ q

b) It is not below freezing and it is not snowingAns.) ~ p Λ ~ q

SLIDE # 10

Q8: Let p and q and r be proposition p:you have the flu .q: you miss the final examination .r: you pass the course.a) ( p → ~r ) ѵ ( q → ~ r)Ans.) if you have flu then you not pass the course or if you miss the final examination then you not pass the course

b) q → ~ rAns.) if you miss the final examination then you not pass the course .

SLIDE # 11

Q9:Let p and q be proposition .p: you drive over 65 miles per hour. you get a speeding ticket write logical connectionDriving over 65 miles per hour is sufficient for getting a speeding ticket .Ans. ) p → q

b) You do not drive over 65 miles per hour.Ans.) ~ p

c) If you do not drive over 65 miles per hour then you will not getting a speeding ticket. Ans.) ~ p → ~ q

SLIDE # 12

Q23: State the converse , contra positive , and inverse of each of these conditional statements

1)If it snows today , I will ski tomorrow.Ans.) converse:If I will ski tomorrow then it snows today Contra positive:If I will not ski tomorrow then it not snow todayInverse:If it not snow today then I will not ski tomorrow

SLIDE # 13

2) I come to class whenever there is going to be a quiz .

Ans.) converse:If there is going to be a quiz then I come to classContra positive:If there is not going to be a quiz then I not come to class .Inverse :If I not come to class then there is not going to be a quiz

Q24: state the converse contra positive and inverse.

SLIDE # 14

Of each of these conditional statement a) If it snows tonight then I will stay at home Ans. ) converse :If I will stay at home then it snows tonight Contra positive:If I will not stay at home then it not snow tonight Inverse : If it not snow tonight then I will not stay at home

b) I go to beach whenever it is a sunny summer day.

SLIDE # 15

Converse : If it is a sunny summer day then I go to beach

Contra positive:if it is not a sunny summer day then I not go to

beachInverse:If I not go to beach then it is not sunny summer day

SLIDE # 16

Q25: How many rows appear in a truth table for each of these compound proposition

a) p → ~ pAns.) number of rows =2

b) (p ѵ ~ r) Λ (q ѵ ~ s)Ans.) number of rows =16

c) q ѵ p ѵ ~ s ѵ ~r ѵ ~ t ѵ uAns.) number of rows = 64

SLIDE # 17

d) (p Λ r Λ t) ↔ ( q Λ t)Ans.) number of rows =16

Q26: How many rows appear in a truth tablea) (q → ~ p)Ans.) number of rows =4

b) ( p ѵ ~ t) Λ (p ѵ ~ s)Ans.) number of rows=8

c) ( p Λ r Λ s ) ѵ ( q Λ t) ѵ (r Λ ~ t)

Ans.) number of rows = 32

SLIDE # 18

Q 27: Construct a truth table for each of these compound proposition

a) p Λ ~ p

b) p ѵ ~ p

P ~p P Λ ~P

T F F

F T F

p ~ p P ѵ ~p

T F T

F T T

SLIDE # 19

c) ( p ѵ ~q ) → q

p q ~q P ѵ ~q

(p ѵ ~q )→ q

T T F T T

T F T T F

F T F F T

F F T T F

SLIDE # 20

Q28: Construct truth table of following.a) p → ~ p

b) p ↔ ~p

p ~p P → ~p

T F F

F T T

p ~p P ↔ ~p

T F F

F T F

SLIDE # 21

C) p + (p ѵ q )

29 :Construct truth tablea) (p ѵ q ) → (p + q)

p q P ѵ q P + ( p ѵ q)

T T T F

T F T F

F T T T

F F F F

p q P ѵ q p+q (p ѵ q)→(p+q)

T T T F F

T F T T T

F T T T T

F F F F T

SLIDE # 22

b) (p + q) → (p Λ q)

p q P+ q P Λ q

P + q → (p q)

T T F T T

T F T F F

F T T F F

F F F F T

SLIDE # 23

C) (p ѵ q ) + (p Λ q)

p q P ѵ q P Λ q (p ѵ q) (p Λ q)

T T T T F

T F T F T

F T T F T

F F F F F

SLIDE # 24

Q30: Construct truth tablea) p + ~p

b) p + ~q

p ~p p+ ~p

T F T

F T T

p q ~q P + ~q

T T F T

T F T F

F T F F

F F T T

SLIDE # 25

C) ~P + ~q

p q ~p ~q ~p +~q

T T F F F

T F F T T

F T T F T

F F T T F