Logical Connectives | Propositional Logic

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-

 

pq q
FFT
FTT
TFF
TTT

 

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-

 

pq q∼p ∨ q
FFTT
FTTT
TFFF
TTTT

 

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-

 

pq q
FFT
FTF
TFF
TTT

 

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-

 

pq q(p ∧ q) ∨ (∼p ∧ ∼q)
FFTT
FTFF
TFFF
TTTT

 

Converting English Sentences To Propositional Logic-

 

While solving questions, the following replacements are very useful-

 

WordReplacement
AndConjunction (∧)
OrDisjunction (∨)
ButAnd
WheneverIf
WhenIf
Either p or qp or q
Neither p nor qNot 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-

  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.
  11. It is false that he is poor or clever but not honest.
  12. It is hot or else it is both cold and cloudy.
  13. I will not go to class unless you come.
  14. We will leave whenever he comes.
  15. Either today is Sunday or Monday.
  16. You will qualify GATE only if you work hard.
  17. Presence of cycle in a single instance RAG is a necessary and sufficient condition for deadlock.
  18. Presence of cycle in a multi instance RAG is a necessary but not sufficient condition for deadlock.
  19. I will dance only if you sing.
  20. 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 ∨ (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?

  1. S1 is correct and S2 is incorrect.
  2. S1 is incorrect and S2 is correct.
  3. Both are correct.
  4. 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)

 q

(Ticket is sufficient for entry)

FFT
FTT
TFF
TTT

 

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)

 q

(Ticket is necessary for entry)

FFT
FTT
TFF
TTT

 

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,

Watch this Video Lecture

 

Next Article- Converse, Inverse and Contrapositive

 

Get more notes and other study material of Propositional Logic.

Summary
Logical Connectives | Propositional Logic
Article Name
Logical Connectives | Propositional Logic
Description
Logical Connectives are used to connect propositions. Properties and Formulas of Conditional and Biconditional. Examples of Propositional Logic. Practice Problems on Converting English Sentences to Propositional Logic.
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Gate Vidyalay
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