Conditional and Bi-conditional connectivity

To understand conditional and bi-conditional connectivity, we should go through the previous section, i.e., Logical connectives in Discrete mathematics.

Logical connectivity

Logical connectivity can be described as the operators that are used to connect one or more than one propositions or predicate logic. The propositional logic is used to contain 5 basic connectives, which are described as follows:

1. Negation
2. Conjunction
3. Disjunction
4. Conditional
5. Bi-conditional

In this section, we will learn about some important formulas, properties, results, and proofs of implication, and bi-implication. We will also learn about converting English sentences into propositional logic.

Conditional Propositions:

The conditional propositional is also known as the implication proposition. Suppose there are two propositions, x and y. The conditional proposition will have the form "if x then y". The conditional proposition is used to have some properties, which are shown below:

• If the two propositions x and y are true or when x is false, in this case, their propositions will be true.
• If x is true and y is false, in this case, their propositions will be false.

Truth table:

The conditional proposition has the following truth table:

Significance of x → y

We can interpret the x → y in the following way:

• If x then y
• X implies y
• X only if y
• X iff y
• X is sufficient for y
• X without y is possible and can exist
• y is necessary for x.
• y follows from x
• x without y is not possible and cannot exist.
• y whenever x

Formulas:

When we have a question related to conditional propositional, then we should remember some points, which are shown below:

• It is possible to replace x → y with ∼x ∨ y.
• x → y is equivalent to ∼y → ∼x.

Proof of Conditional Proposition

The logical equivalence of x → y with ∼x ∨ y can be shown with the help of following table:

Also,

The logical equivalence of x → y with ∼x ∨ y can also be shown with the help of following derivation:

Bi-conditional Propositions

The bi-conditional is also known as the bi-implication proposition. Suppose there are two propositions, x and y. The bi-conditional proposition will have the form "x if and only if y". The bi-conditional proposition is used to have some properties, which are shown below:

• If the two propositions x and y are true or x and y both are false, in this case, their propositions will be true.
• The proposition of x and y will be false in all the other cases.

Truth table:

The bi-conditional proposition has the following truth table:

Significance of x ↔ y

We can interpret the x ↔ y in the following way:

• (If x then q) and (If y then x)
• x if and only if y
• x and y are equivalent
• (x if y) and (y if x)
• ∼x and ∼y are equivalent
• Y if and only if x
• X and y cannot exist without each other
• X is necessary and sufficient for y
• Either x and y both exist, or none of them exist
• Y is necessary and sufficient for x
• X and y are necessary and sufficient for each other

Formulas:

When we have a question related to conditional propositional, then we should remember some points, which are shown below:

• It is possible to replace x ↔ y with (x ∧ y) ∨ (∼x ∧ ∼y).
• x ↔ y is equivalent to EX-NOR Gate.

Proof of Conditional Proposition

The logical equivalence of x ↔y and (x ∧ y) ∨ (∼x ∧ ∼y) can be shown with the help of following table:

Converting English sentences into Propositional Logic

When we try to solve the questions, there it will be needed some replacements, which are shown below:

Now we will understand the problem while converting English sentences with the help of solving some problems, which are shown below.

Examples of Problems based on Converting English sentence

Here we will write the English sentence in the symbolic form. These sentences are shown below:

1. If it is sunny outside, then I will go to college.

Solution:

We have the following details:

The given statement is, "If it is sunny outside, then I will go to college."

This statement must have the form: "if x then y".

So, this statement contains a symbolic form, i.e., x → y, where

X: It is sunny outside

Y: I will go to college

2. If I study hard, then I will get good marks on the exam.

Solution:

We have the following details:

The given statement is, "If I study hard, then I will get good marks on the exam."

This statement must have the form: "if x then y".

So, this statement contains a symbolic form, i.e., x → y, where

X: I study hard

Y: I will get good marks on the exam

3. He is hard working but does not get good marks.

Solution:

We have the following details:

The given statement is "He is hard working but does not get good marks."

In this statement, "but" can be replaced with "and".

After replacing the sentence will be - "He is hard working and does not get good marks".

So, this statement contains a symbolic form, i.e., x ∧ y, where

X: He is hard working

Y: He does not get good marks

4. If x = y and y = z, then x = z.

Solution:

We have the following details:

The given statement is, "If x = y and y = z, then x = z."

This statement must have the form: "if x then y".

So, this statement contains a symbolic form, i.e., (x ∧ y) → z, where

x: x = y

y: y = z

z: x = z

5. Neither Harry nor Jack is responsible for this mistake

Solution:

We have the following details:

The given statement is, "Neither Harry nor Jack is responsible for this mistake."

This statement must have the form: "Neither x nor y".

We can rewrite Neither x nor y in another way, i.e., Not x and Not y.

So, this statement contains a symbolic form, i.e., ∼x ∧ ∼y, where

X: Harry is responsible for this mistake

Y: Jack is responsible for this mistake

6. I will go to the beach if and only if it is sunny.

Solution:

We have the following details:

The given statement is, "I will go to the beach if and only if it is sunny."

This statement must have the form: "x if and only if y".

So, this statement contains a symbolic form, i.e., x ↔ y, where

X: I will go to the beach

Y: If it is sunny

7. I will go to the office if and only if jack is coming.

Solution:

We have the following details:

The given statement is, "I will go to the office if and only if jack is coming."

This statement must have the form: "x if and only if y".

So, this statement contains a symbolic form, i.e., x ↔ y, where

X: I will go to the office

Y: If jack is coming

8. I will stay only if it is raining outside.

Solution:

We have the following details:

The given statement is, "I will stay only if it is raining outside."

This statement must have the form: "x only if y".

So, this statement contains a symbolic form, i.e., x → y, where

X: I will stay

Y: It is raining outside

9. I will stay if it is raining outside

Solution:

We have the following details:

The given statement is, "I will stay if it is raining outside."

This statement must have the form: "y if x".

So, this statement contains a symbolic form, i.e., x → y, where

X: It is raining outside

Y: I will stay

10. It is false that he is hard-working but does not get good marks.

Solution:

We have the following details:

The given statement is, "It is false that he is hard working but does not get good marks."

In this statement, "but" can be replaced with "and".

After replacing the sentence will be - "It is false that he is hard working and does not get good marks".

So, this statement contains a symbolic form, i.e., ∼(x ∧ ∼y), where

X: He is hard working

Y: He gets good marks

11. It is false that he is hard-working or intelligent but does not get good marks.

Solution:

We have the following details:

The given statement is, "It is false that he is hard working or intelligent but does not get good marks."

In this statement, "but" can be replaced with "and".

After replacing the sentence will be - "It is false that he is hard working and does not get good marks".

So, this statement contains a symbolic form, i.e., ∼((x ∨ y) ∧ ∼z), where

X: He is hard working

Y: He is intelligent

Z: He gets good marks

12. I will not do my word unless my friends are coming.

Solution:

We have the following details:

The given statement is, "I will not do my word unless my friends are coming."

This statement must have the form: "x unless y".

So, this statement contains a symbolic form, i.e., ∼y → x, where

X: I will do my work

Y: My friends are coming

13. I will go to college whenever my friends come.

Solution:

We have the following details:

The given statement is, "I will go to college whenever my friends come."

In this statement, "whenever" can be replaced with "if".

After replacing the sentence will be - "I will go to college if my friends come."

This statement must have the form: "y if x".

So, this statement contains a symbolic form, i.e., x → y, where

X: My friends come

Y: I will go to college

14. Either you leave, or I will tell my parents.

Solution:

We have the following details:

The given statement is, "Either you leave, or I will tell my parents."

We can rewrite this statement as "You leave, or I will tell my parents".

So, this statement contains a symbolic form, i.e., x ∨ y, where

X: You leave

Y: I will tell my parents

15. You will earn grade A only if you study hard.

Solution:

We have the following details:

The given statement is "You will earn grade A only if you study hard."

This statement must have the form: "x only if y".

So, this statement contains a symbolic form, i.e., x → y, where

X: You will earn grade A

Y: You study hard

16. Air is a necessary and sufficient condition for human life.

Solution:

We have the following details:

The given statement is, "Air is a necessary and sufficient condition for human life."

This statement must have the form: "x is necessary and sufficient for y".

So, this statement contains a symbolic form, i.e., x ↔ y, where

X: Air

Y: Human life

17. A table having four sides is a necessary but not sufficient condition for its being square.

Solution:

We have the following details:

The given statement is "A table having four sides is a necessary but not sufficient condition for its being square."

This statement must have the form: "x is necessary but not sufficient for y".

So, this statement contains a symbolic form, i.e., (y → x) ∧ ∼(x → y), where

X: A table having four sides

Y: Its being square

18. I will take my umbrella with me only if there is raining outside.

Solution:

We have the following details:

The given statement is, "I will take my umbrella with me only if there is raining outside."

This statement must have the form: "x only if y".

So, this statement contains a symbolic form, i.e., x → y, where

X: I will take my umbrella with me

Y: There is raining outside

19. Neither Jack nor his girlfriend talks about his wedding.

Solution:

We have the following details:

The given statement is "Neither Jack nor his girlfriend talks about his weeding."

This statement must have the form: "Neither x nor y".

We can rewrite this statement in the form: "Not x and not y".

So, this statement contains a symbolic form, i.e., ∼x ∧ ∼y, where

X: Jack talks about his wedding

Y: His girlfriend talks about his wedding

Example of Logical connectivity

Example 1: In this example, we have two statements, S1 and S2, where

Statement 1 (S1): 90% marks are sufficient to clear the cut-off list.

Statement 2 (S2): 90% marks are necessary to clear the cut-off list.

Now we have to determine which statements 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:

S1: 90% marks are sufficient to clear the cut-off list.

The above statement contains a form, i.e., "x is sufficient for y", where

X: You get 90% marks

Y: You can clear the cut-off list.

So, this statement contains a symbolic form, i.e., x → y, where

The x → y will be held if it contains the following truth table:

Here,

• According to Row 2, it is possible that you get 90% marks and you can clear the cut-off list.
• However, it is impossible to clear the cut-off list without getting 90% marks.
• According to Row 3, it is not possible that you get 90% marks, and you do not clear the cut-off list.
• However, there can be a possible case in which you get 90% marks but do not clear the cut-off list.
• So, the above described truth table will not hold.

Hence, the statement S1- "90% marks are sufficient to clear the cut-off list" is logically incorrect.

S2: 90% marks are necessary to clear the cut-off list.

The above statement contains a form, i.e., "x is necessary for y", where

X: You can clear the cut-off list

Y: You get 90% marks

So, this statement contains a symbolic form, i.e., x → y, where

The x → y will be held if it contains the following truth table:

In the above truth table, all the rows make the correct sense.

Hence the statement S2- "90% marks are necessary to clear the cut-off list" is logically correct"

Thus, Option (B) is correct.