Logical Connectives-
Before you go through this article, make sure that you have gone through the previous article on Logical Connectives.
We have discussed-
- Logical connectives are the operators used to combine one or more propositions.
- In propositional logic. there are 5 basic connectives-
In this article, we will discuss-
- Some important results, properties and formulas of conditional and biconditional.
- Converting English sentences to propositional logic.
Conditional-
If p and q are two propositions, then-
Proposition of the type “If p then q” is called a conditional or implication proposition. |
- It is true when both p and q are true or when p is false.
- It is false when p is true and q is false.
Also Read- Propositions
Truth Table-
p | q | p → q |
F | F | T |
F | T | T |
T | F | F |
T | T | T |
Significance Of P→Q:
p → q may be interpreted as-
- If p then q
- p implies q
- q follows from p
- q if p
- q whenever p
- p only if q
- p is sufficient for q
- q is necessary for p
- q without p is possible and can exist
- p without q is impossible and can not exist
Formulas-
While solving questions, remember-
- You can always replace p → q with ∼p ∨ q.
- p → q is equivalent to ∼q → ∼p.
Proof-
The following table clearly shows that p → q and ∼p ∨ q are logically equivalent-
p | q | p → q | ∼p ∨ q |
F | F | T | T |
F | T | T | T |
T | F | F | F |
T | T | T | T |
Also,
The following derivation shows that p → q and ∼q → ∼p are logically equivalent-
∼q → ∼p
= ∼(∼q) ∨ ∼p
= q ∨ ∼p
= ∼p ∨ q
Biconditional-
If p and q are two propositions, then-
Proposition of the type “p if and only if q” is called a biconditional or bi-implication proposition. |
- It is true when either both p and q are true or both p and q are false.
- It is false in all other cases.
Truth Table-
p | q | p ↔ q |
F | F | T |
F | T | F |
T | F | F |
T | T | T |
Significance Of P↔Q:
p ↔ q may be interpreted as-
- (If p then q) and (If q then p)
- p if and only if q
- q if and only if p
- (p if q) and (q if p)
- p is necessary and sufficient for q
- q is necessary and sufficient for p
- p and q are necessary and sufficient for each other
- p and q can not exist without each other
- Either p and q both exist or none of them exist
- p and q are equivalent
- ∼p and ∼q are equivalent
Formulas-
While solving questions, remember-
- Biconditional is equivalent to EX-NOR Gate.
- You can always replace p ↔ q with (p ∧ q) ∨ (∼p ∧ ∼q).
Proof-
The following table clearly shows that p ↔ q and (p ∧ q) ∨ (∼p ∧ ∼q) are logically equivalent-
p | q | p → q | (p ∧ q) ∨ (∼p ∧ ∼q) |
F | F | T | T |
F | T | F | F |
T | F | F | F |
T | T | T | T |
Converting English Sentences To Propositional Logic-
While solving questions, the following replacements are very useful-
Word | Replacement |
And | Conjunction (∧) |
Or | Disjunction (∨) |
But | And |
Whenever | If |
When | If |
Either p or q | p or q |
Neither p nor q | Not p and Not q |
p unless q | ∼q → p |
p is necessary but not sufficient for q | (q → p) and ∼(p → q) |
To understand better, let us try solving the following problems.
PRACTICE PROBLEMS BASED ON CONVERTING ENGLISH SENTENCES-
Problem-01:
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.
- It is false that he is poor or clever but not honest.
- It is hot or else it is both cold and cloudy.
- I will not go to class unless you come.
- We will leave whenever he comes.
- Either today is Sunday or Monday.
- You will qualify GATE only if you work hard.
- Presence of cycle in a single instance RAG is a necessary and sufficient condition for deadlock.
- Presence of cycle in a multi instance RAG is a necessary but not sufficient condition for deadlock.
- I will dance only if you sing.
- Neither the red nor the green is available in size 5.
Solution-
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
Part-11:
We have-
- The given sentence is- “It is false that he is poor or clever but not honest.”
- We can replace “but” with “and”.
- Then, the sentence is- “It is false that he is poor or clever and not honest.”
So, the symbolic form is ∼((p ∨ q) ∧ ∼r) where-
p : He is poor
q : He is clever
r : He is honest
Part-12:
We have-
- The given sentence is- “It is hot or else it is both cold and cloudy.”
- It can be re-written as- “It is hot or it is both cold and cloudy.”
So, the symbolic form is p ∨ (q ∧ r) where-
p : It is hot
q : It is cold
r : It is cloudy
Part-13:
We have-
- The given sentence is- “I will not go to class unless you come.”
- This sentence is of the form- “p unless q”.
So, the symbolic form is ∼ q → p where-
p : I will go to class
q : You come
Part-14:
We have-
- The given sentence is- “We will leave whenever he comes.”
- We can replace “whenever” with “if”.
- Then, the sentence is- “We will leave if he comes.”
- This sentence is of the form- “q if p”.
So, the symbolic form is p → q where-
p : He comes
q : We will leave
Part-15:
We have-
- The given sentence is- “Either today is Sunday or Monday.”
- It can be re-written as- “Today is Sunday or Monday.”
So, the symbolic form is p ∨ q where-
p : Today is Sunday
q : Today is Monday
Part-16:
We have-
- The given sentence is- “You will qualify GATE only if you work hard.”
- This sentence is of the form- “p only if q”.
So, the symbolic form is p → q where-
p : You will qualify GATE
q : You work hard
Part-17:
We have-
- The given sentence is- “Presence of cycle in a single instance RAG is a necessary and sufficient condition for deadlock.”
- This sentence is of the form- “p is necessary and sufficient for q”.
So, the symbolic form is p ↔ q where-
p : Presence of cycle in a single instance RAG
q : Presence of deadlock
Part-18:
We have-
- The given sentence is- “Presence of cycle in a multi instance RAG is a necessary but not sufficient condition for deadlock.”
- This sentence is of the form- “p is necessary but not sufficient for q”.
So, the symbolic form is (q → p) ∧ ∼(p → q) where-
p : Presence of cycle in a multi instance RAG
q : Presence of deadlock
Part-19:
We have-
- The given sentence is- “I will dance only if you sing.”
- This sentence is of the form- “p only if q”.
So, the symbolic form is p → q where-
p : I will dance
q : You sing
Part-20:
We have-
- The given sentence is- “Neither the red nor the green is available in size 5.”
- This sentence is of the form- “Neither p nor q”.
- “Neither p nor q” can be written as “Not p and Not q”.
So, the symbolic form is ∼p ∧ ∼q where-
p : Red is available in size 5
q : Green is available in size 5
Problem-02:
Consider the following two statements-
S1 : Ticket is sufficient to enter movie theater.
S2 : Ticket is necessary to enter movie theater.
Which of the statements is/ are logically correct?
- S1 is correct and S2 is incorrect.
- S1 is incorrect and S2 is correct.
- Both are correct.
- Both are incorrect.
Solution-
Statement S1 : Ticket is Sufficient To Enter Movie Theater-
This statement is of the form- “p is sufficient for q” where-
p : You have a ticket
q : You can enter a movie theater
So, the symbolic form is p → q
For p → q to hold, its truth table must hold-
p
(Ticket) |
q
(Entry) |
p → q
(Ticket is sufficient for entry) |
F | F | T |
F | T | T |
T | F | F |
T | T | T |
Here,
- Row-2 states it is possible that you do not have a ticket and you can enter the theater.
- However, it is not possible to enter a movie theater without ticket.
- Row-3 states it is not possible that you have a ticket and you do not enter the theater.
- However, there might be a case possible when you have a ticket but do not enter the theater.
- So, the truth table does not hold.
Thus, the statement- “Ticket is sufficient for entry” is logically incorrect.
Statement S2 : Ticket is Necessary To Enter Movie Theater-
This statement is of the form- “q is necessary for p” where-
p : You can enter a movie theater
q : You have a ticket
So, the symbolic form is p → q
For p → q to hold, its truth table must hold-
p
(Entry) |
q
(Ticket) |
p → q
(Ticket is necessary for entry) |
F | F | T |
F | T | T |
T | F | F |
T | T | T |
Here, All the rows of the truth table make the correct sense.
Thus, the statement- “Ticket is necessary for entry” is logically correct.
Thus, Option (B) is correct.
To gain better understanding about converting English sentences,
Next Article- Converse, Inverse and Contrapositive
Get more notes and other study material of Propositional Logic.