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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 # 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