Module 3

Q. 1

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.

Solutions-

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)

Q.II 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.

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

Module 3

Q. 1

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.

Solutions-

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)

Q.II 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.

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

Module 3

Q. 1

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.

Solutions-

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)

Q.II 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.

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