## 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-

• 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.

### 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.

## 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 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

### 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

### 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) 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,

Watch this Video Lecture

Next Article- Converse, Inverse and Contrapositive

Get more notes and other study material of Propositional Logic.

## Propositions-

Before you go through this article, make sure that you have gone through the previous article on Propositions.

We have discussed-

• Proposition is a declarative statement that is either true or false but not both.
• Connectives are used to combine the propositions.

## Logical Connectives-

 Connectives are the operators that are used to combine one or more propositions.

In propositional logic, there are 5 basic connectives-

 Name of Connective Connective Word Symbol Negation Not ⌉ or ∼ or ‘ or – Conjunction And ∧ Disjunction Or ∨ Conditional If-then → Biconditional If and only if ↔

## 1. Negation-

If p is a proposition, then negation of p is a proposition which is-

• True when p is false
• False when p is true.

### Truth Table-

 p ∼p F T T F

### Example-

If p : It is raining outside.

Then, Negation of p is-

∼p : It is not raining outside.

## 2. Conjunction-

If p and q are two propositions, then conjunction of p and q is a proposition which is-

• True when both p and q are true
• False when both p and q are false

### Truth Table-

 p q p ∧ q F F F F T F T F F T T T

### Example-

If p and q are two propositions where-

• p : 2 + 4 = 6
• q : It is raining outside.

Then, conjunction of p and q is-

p ∧ q : 2 + 4 = 6 and it is raining outside

## 3. Disjunction-

If p and q are two propositions, then disjunction of p and q is a proposition which is-

• True when either one of p or q or both are true
• False when both p and q are false

### Truth Table-

 p q p ∨ q F F F F T T T F T T T T

### Example-

If p and q are two propositions where-

• p : 2 + 4 = 6
• q : It is raining outside

Then, disjunction of p and q is-

p ∨ q : 2 + 4 = 6 or it is raining outside

## 4. 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.

### Truth Table-

 p q p → q F F T F T T T F F T T T

### Examples-

• If a = b and b = c then a = c.
• If I will go to Australia, then I will earn more money.

## 5. 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

### Examples-

• He goes to play a match if and only if it does not rain.
• Birds fly if and only if sky is clear.

## Note-01:

• Negation ≡ NOT Gate of digital electronics.
• Conjunction ≡ AND Gate of digital electronics.
• Disjunction ≡ OR Gate of digital electronics.
• Biconditional = EX-NOR Gate of digital electronics.

## Note-02:

• Each logical connective has some priority.
• This priority order is important while solving questions.
• The decreasing order of priority is-

## Note-03:

• Negation, Conjunction, Disjunction and Biconditional are both commutative and associative.
• Conditional is neither commutative nor associative.

To gain better understanding about Logical Connectives,

Watch this Video Lecture

Next Article- Converting English Sentences To Propositional Logic

Get more notes and other study material of Propositional Logic.