In propositional calculus we do calculations on propositions,with well defined symbols,to determine the truth or falsehood of compound propositions.

Answer to the problems are provided in the doc file given above

PROPOSITIONAL CALCULUS

There are many clouds in the sky but it did not rain

P: There are many clouds in the sky

q: It rain

then__________________ (1)

P v q

P ^ q

P v (~q)

P ^ (~q)

I will get first class if and only if I study well and score above 80 in mathematics

P: I will get first class

q: I study well

r: score above 80 in mathematics

then_________ (1)

P ↔(q ^ r)

(P ^ q)↔ r

(P ^ r)↔ q

q ↔ p

Computers are cheap but Softwares are costly.

P: Computers are cheap

q:Softwares are costly.

Then____________ (1)

P v q

P ^ q

P v (~q)

P ^ (~q)

It is very hot and humid or Ramesh is having heart problem.

P: It is very hot

q:It is very humid

r:Ramesh is having heart problem.

Then **__** (1)

P ^ q ^ r

(p v q) ^ r

(P ^ q) v r

P v q v r

.In small restaurants the food is good and service is poor.

P:In small restaurants the food is good

q: service is poor

then ** _** (1)

P ^ q

P v q

(~p) ^ q

P ^ (~q)

If I finish my submission before 5:00 pm in the evening and it is not very hot, I will go

and play a game of hockey.

P:I finish my submission before 5:00 pm

q: It is very hot

r: I will go

s: I will play a game of hockey.

Then___________ (1)

(p ^ q) →(r ^ s)

[P ^ (~q)] →(r ^ s)

(p ^ q) →(r v s)

[P ^ (~q)] →(r v s)

If p denote the statement, The material is interesting.

q denote the statement, The exercises are challenging.

r denote the statement, The course is enjoyable.

Write the statement in symbolic form.

The material is interesting and exercises are challenging,Then__________ (1)

p v r

p ^ r

p ^ q

q ^ r

Either the material is interesting or exercises are not challenging . (1)

p v q

p v (~ q)

~p v q

~p v ~q

The material is interesting means exercises are challenging and conversely. (1)

p → q

q → p

(p → q) ^ (q → p)

(p → q) v (q → p)

If the material is not interesting and exercises are not challenging, then

the course is not enjoyable. (2)

(p ^ q)→ r

(~p ^ ~q) → r

(~p ^ ~q) →~ r

(p ^ q) →~ r

~(p v q) is_____________ (1)

(~p) v q

(~p) v (~q)

(~p) ^ (~q)

p v (~q)

~ (p ^ q) is_____________ (1)

(~p) v q

(~p) v (~q)

(~p) ^ (~q)

p v (~q)

p v (q ^ r) is______________ (1)

(p v q) ^ (p v r)

(p v q)^ r

(p ^ q)v r

(p ^ q)v (p ^ r)

p ^ (q v r) is______________ (1)

(p v q) ^ (p v r)

(p v q)^ r

(p ^ q)v r

(p ^ q) v (p ^ r)

p v (p ^ q) is______________ (1)

(p ^ q)

(p v q)

P

q

p ^ (p v q) is______________ (1)

(p ^ q)

(p v q)

P

q

p → q is logically equivalent to___________ (2)

(p ^ q)

(p v q)

(~p) v q

P v (~q)

```
p ↔ q is logically equivalent to___________ (2)
[(~p) v q] ^ [(~q) v p]
[(~p) ^ q] v [(~q) ^ p]
[(~p) v q] v [(~q) v p]
[(~p) ^ q] ^ [(~q) ^ p]
For p v (~p), p ^ (~p) (2)
Both true
Both false
p v (~p) is true and p ^ (~p) is false
p v (~p) is false and p ^ (~p) is true
p v (~p ^ q) is_________________ (2)
p v q
p ^ q
0
1
p ^ (~p v q) is_________________ (2)
p v q
p ^ q
0
1
P → (q →p) is _____________ (4)
Tautology
Contradiction
Contingency
None of these
(p ^ q) → p is _______________ (4)
Tautology
Contradiction
Contingency
None of these
(~p) v q → q is _______________ (4)
Tautology
Contradiction
Contingency
None of these
```

If ‘p’ stands for ‘I run fast’ and ‘q’ stands for ‘ I shall win’,write the symbolic form of

```
I do not run fast. (1)
p
q
~p
~q
If I run fast, I shall win. (1)
p → q
q → p
p ↔ q
~p →~q
I run fast or I shall not win. (1)
p v q
p ^ q
p v ~q
~p v q
I run fast and I shall win. (1)
p v q
p ^ q
p → q
q → p
I shall win if and only if I run fast. (1)
p → q
q → p
q ↔ p
~p → q
I neither run fast nor I shall win. (1)
~p ^ ~ q
~p v ~ q
p ^ ~ q
~p ^ q
n^3+2n for n ≥1 is divisible by __________ (1)
2
3
4
6
n^4-4n^2 for n ≥2 is divisible by _________ (1)
2
3
4
6
1 + 4 + 7 + ……………… + (3n – 2 ) is ___________ (4)
n(3n – 1 )/2
n(3n + 1 )/2
3n(n + 1)/2
3n(n – 1 )/2
1.2 + 2.3 + 3.4 + …………………. + n(n + 1) is __________ (4)
n(n + 1)/2
n(n + 1)(n + 2)/3
n(n – 1 )(n – 2 )/3
n(n + 2)/2
1^2+3^2+5^2+⋯……….+ 〖(2n-1)〗^2 is __________ (4)
n(n – 1 )(n – 2 )/3
n(2n – 1 )(2n + 1 )/3
n(2n + 1 )(2n + 3 )/3
n(n + 2)/2
1^3+2^3+3^3+⋯……….+ n^3 is __________ (4)
n(n – 1 )(n – 2 )/3
n(n + 1 )(n + 2 )/3
〖(1+2+⋯…….+n)〗^2
n(n + 2)/2
( p v q) ^ ( p → r) ^ ( q → s) is equivalent to___________ (4)
s ^ r
s → r
s v r
none of these
```