Unit - 3
Propositional Logic
Q1) Identify which of the following statements are propositions-
- France is a country.
- 2020 will be a leap year.
- Sun rises in the west.
- P(x) : x + 6 = 7
- P(5) : 5 + 6 = 2
- Apples are oranges.
- Grapes are black.
- Two and two makes 4.
- x > 10
- Open the door.
- Are you tired?
- What a bright sunny day!
- Mumbai is in India.
- I always tell truth.
- I always tell lie.
- Do not go there.
- This sentence is true.
- This sentence is false.
- It will rain tomorrow.
- Fan is rotating.
A1)
- Proposition (True)
- Proposition (True)
- Proposition (False)
- Not a proposition (Predicate)
- Proposition (False)
- Proposition (False)
- Proposition (False)
- Proposition (True)
- Not a proposition (Predicate)
- Not a proposition (Command)
- Not a proposition (Question)
- Not a proposition (Exclamation)
- Proposition (True)
- Proposition (True)
- Not a proposition (Inconsistent)
- Not a proposition (Command)
- Proposition (True)
- Not a proposition (Inconsistent)
- Proposition (Will be confirmed tomorrow whether true or false)
- Proposition (True if fan is rotating otherwise false)
Q2) Write the following English sentences in symbolic form-
- If it rains, then I will stay at home.
- If I will go to Australia, then I will earn more money.
- He is poor but honest.
- If a = b and b = c then a = c.
- Neither it is hot nor cold today.
- He goes to play a match if and only if it does not rain.
- Birds fly if and only if sky is clear.
- I will go only if he stays.
- I will go if he stays.
- 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