Unit - 3

Propositional Logic

Q1) Identify which of the following statements are propositions-

1. France is a country.
2. 2020 will be a leap year.
3. Sun rises in the west.
4. P(x) : x + 6 = 7
5. P(5) : 5 + 6 = 2
6. Apples are oranges.
7. Grapes are black.
8. Two and two makes 4.
9. x > 10
10. Open the door.
11. Are you tired?
12. What a bright sunny day!
13. Mumbai is in India.
14. I always tell truth.
15. I always tell lie.
16. Do not go there.
17. This sentence is true.
18. This sentence is false.
19. It will rain tomorrow.
20. Fan is rotating.

A1) 

1. Proposition (True)
2. Proposition (True)
3. Proposition (False)
4. Not a proposition (Predicate)
5. Proposition (False)
6. Proposition (False)
7. Proposition (False)
8. Proposition (True)
9. Not a proposition (Predicate)
10. Not a proposition (Command)
11. Not a proposition (Question)
12. Not a proposition (Exclamation)
13. Proposition (True)
14. Proposition (True)
15. Not a proposition (Inconsistent)
16. Not a proposition (Command)
17. Proposition (True)
18. Not a proposition (Inconsistent)
19. Proposition (Will be confirmed tomorrow whether true or false)
20. Proposition (True if fan is rotating otherwise false)

Q2) Write the following English sentences in symbolic form-

1. If it rains, then I will stay at home.
2. If I will go to Australia, then I will earn more money.
3. He is poor but honest.
4. If a = b and b = c then a = c.
5. Neither it is hot nor cold today.
6. He goes to play a match if and only if it does not rain.
7. Birds fly if and only if sky is clear.
8. I will go only if he stays.
9. I will go if he stays.
10. It is false that he is poor but not honest.

A2)

Part-01:

We have-

• The given sentence is- “If it rains, then I will stay at home.”
• This sentence is of the form- “If p then q”.

So, the symbolic form is p → q where-

p : It rains

q : I will stay at home

Part-02:

We have-

• The given sentence is- “If I will go to Australia, then I will earn more money.”
• This sentence is of the form- “If p then q”.

So, the symbolic form is p → q where-

p : I will go to Australia

q : I will earn more money

Part-03:

We have-

• The given sentence is- “He is poor but honest.”
• We can replace “but” with “and”.
• Then, the sentence is- “He is poor and honest.”

So, the symbolic form is p ∧ q where-

p : He is poor

q : He is honest

Part-04:

We have-

• The given sentence is- “If a = b and b = c then a = c.”
• This sentence is of the form- “If p then q”.

So, the symbolic form is (p ∧ q) → r where-

p : a = b

q : b = c

r : a = c

Part-05:

We have-

• The given sentence is- “Neither it is hot nor cold today.”
• This sentence is of the form- “Neither p nor q”.
• “Neither p nor q” can be re-written as “Not p and Not q”.

So, the symbolic form is ∼p ∧ ∼q where-

p : It is hot today

q : It is cold today

Part-06:

We have-

• The given sentence is- “He goes to play a match if and only if it does not rain.”
• This sentence is of the form- “p if and only if q”.

So, the symbolic form is p ↔ q where-

p : He goes to play a match

q : It does not rain

Part-07:

We have-

• The given sentence is- “Birds fly if and only if sky is clear.”
• This sentence is of the form- “p if and only if q”.

So, the symbolic form is p ↔ q where-

p : Birds fly

q : Sky is clear

Part-08:

We have-

• The given sentence is- “I will go only if he stays.”
• This sentence is of the form- “p only if q”.

So, the symbolic form is p → q where-

p : I will go

q : He stays

Part-09:

We have-

• The given sentence is- “I will go if he stays.”
• This sentence is of the form- “q if p”.

So, the symbolic form is p → q where-

p : He stays

q : I will go

Part-10:

We have-

• The given sentence is- “It is false that he is poor but not honest.”
• We can replace “but” with “and”.
• Then, the sentence is- “It is false that he is poor and not honest.”

So, the symbolic form is ∼ (p ∧ ∼q) where-

p : He is poor

q : He is honest