1. DISCRETE STRUCTURE
Bs (IT)

2. Discrete structure d) 5+7=10 e) x+2=11
Question no 1
What are truth value of those that are proposition ?
Boston is a capital of Massachusetts.
Ans.) proposition T
b) Miami is capital of Florida.
Ans.) proposition T
2+3=5
ans.) proposition T
d) 5+7=10
ans.) proposition .F
e) x+2=11
no proposition

3. Slide # 1 Ans.) proposition. There are black flies in Maine
f) Answer this question .
Ans.) no proposition
Q2: Which of these are proposition ? What are the truth value of these that are proposition?
a: Do not pass go.
b: What time is it .
Ans.) no proposition
c: There are no black flies in Maine.
Ans.) proposition. There are black flies in Maine

4. e: The moon is made of green cheese. Ans.) proposition f: 2n ≥ 100
d: 4+x=5
Ans.) no proposition
e: The moon is made of green cheese.
Ans.) proposition
f: 2n ≥ 100
Ans.) no proposition
Q3:What is negation of each of these proposition?
Slide # 2

5. Slide # 3 Today is Thursday. b) There is no pollution in new jersey
Ans.) Today is not Thursday
b) There is no pollution in new jersey
Ans.) There is pollution in new jersey
2+1=3
Ans.) is not equal to 3
d) The summer in Maine is hot and sunny
Ans.) The summer in Maine is hot and sunny

6. 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:~ p
Ans.) I not bought a lottery ticket this week
2: p ν q
Ans.) I bought a lottery ticket this week or I won the million dollar jackpot on Friday
Slide # 4

7. Slide # 5 c) P → q p Λ q e)p ↔ q ~p ↔ q
Ans.) If I bought a lottery this week then I won million dollar
jackpot on Friday
p Λ q
Ans.) 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
~p ↔ q

8. Ans.) if I not bought a lottery ticket this week then I not win million dollar jackpot on Friday
g) ~ p Λ ~q
Ans.) 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 # 6

9. Slide # 7 ~ p b) p Λ q p → ~ q P ↔ ~q
Ans.) Swimming at new jersey shore is not allowed
b) p Λ q
Ans.) swimming at new jersey shore is allowed and sharks have been spotted near shore
p → ~ q
Ans.) if swimming at new jersey shore is allowed the sharks have not been spotted near shore
P ↔ ~q
Ans.) swimming at new jersey shore is allowed if and only if the sharks have not been spotted near shore

10. 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
q → p
Ans.) if the votes have been counted then the election
is decided.
b) p ↔ q
Ans.) the election is decided if and only if votes have
been counted

11. 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.
It is below freezing and snowing .
Ans.) p Λ q
b) It is not below freezing and it is not snowing
Ans.) ~ p Λ ~ q

12. 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.
( 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 → ~ r
Ans.) if you miss the final examination then you not
pass the course .
Slide # 10

13. Slide # 11 Q9:Let p and q be proposition .p: you drive over
miles per hour. you get a speeding ticket write
logical connection
Driving 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

14. 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 today
Inverse:
If it not snow today then I will not ski tomorrow

15. 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 class
Contra 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.

16. Slide # 14 Of each of these conditional statement
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.

17. Slide # 15 Converse : Contra positive:
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 beach
Inverse:
If I not go to beach then it is not sunny summer day

18. Q25: How many rows appear in a truth table for each of these compound proposition
Ans.) number of rows =2
(p ѵ ~ r) Λ (q ѵ ~ s)
Ans.) number of rows =16
c) q ѵ p ѵ ~ s ѵ ~r ѵ ~ t ѵ u
Ans.) number of rows = 64
Slide # 16

19. Q26: How many rows appear in a truth table (q → ~ p)
d) (p Λ r Λ t) ↔ ( q Λ t)
Ans.) number of rows =16
Q26: How many rows appear in a truth table
(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 # 17

20. Q 27: Construct a truth table for each of these compound proposition
p Λ ~ p
b) p ѵ ~ p P ~p
P Λ ~P T F p
~ p
P ѵ ~p T F
Slide # 18

21. Slide # 19
c) ( p ѵ ~q ) → q p q ~q
P ѵ ~q
(p ѵ ~q ) → q T F

22. Q28: Construct truth table of following. p → ~ p
b) p ↔ ~p p ~p
P → ~p T F p ~p
P ↔ ~p T F
Slide # 20

23. 29 :Construct truth table a) (p ѵ q ) → (p + q)
p + (p ѵ q )
29 :Construct truth table
a) (p ѵ q ) → (p + q) p q
P ѵ q
P + ( p ѵ q) T F p q
P ѵ q
p+q
(p ѵ q)→(p+q) T F
Slide # 21

24. Slide # 22
b) (p + q) → (p Λ q) p q
P+ q
P Λ q
P + q → (p q) T F

25. Slide # 23
C) (p ѵ q ) + (p Λ q) p q
P ѵ q
P Λ q
(p ѵ q) (p Λ q) T F

26. Q30: Construct truth table p + ~p
p + ~q p ~p
p+ ~p T F p q ~q
P + ~q T F
Slide # 24

27. Slide # 25
C) ~P + ~q p q ~p ~q
~p +~q T F