Biconditional Statement in Discrete Mathematics

The bicondition stands for condition in both directions. Biconditional can be described as another type of necessary implication. Sometimes, the biconditional statements are also known as the bi-implication. The biconditional statements are indicated with the help of a symbol ⇔. In this statement, we don't use the same key that we use in implication, i.e., 'If and then'. In this statement, we use the keyword 'if and only if' so that we can join the premise and conclusion. In the bicondition, the statements will be in the form of {(hypothesis/premise) if and only if (conclusion)}. The biconditional statement can also be written in another way, i.e., {(conclusion) if and only if (premise)}.

The biconditional statements can also be described in other words, and according to this, we can create a biconditional statement with the help of true conditional statements. For this, we have to just remove the "if then" part from the conditional statement, and after that, we have to combine the premise and conclusion and tuck them in the phrase "if and only if".

How to write the conditional statements

The biconditional statements can be written in various ways, which are described as follows:

As we know that this statement is biconditional (conditional in both directions). That's why we also write this statement in the form of a converse statement, which is described as follows:

So we have noticed that it is possible to create two bi-conditional statements. So the conditional statements can be called a one-way street, and the bi-conditional statements can be called a two-way street.

There are two other ways in which we can write the bi-conditional statements, which are described as follows:

P is necessary and sufficient for Q

P iff Q, where 'iff' stands for 'if and only if'

To generate a bi-conditional statement, it is necessary that the conditional statement and the converse statements are true.

Converse statements and Compound statements

The converse and compound statements can be described as biconditional statements. Here we will describe the conditional, converse, and compound statements.

Conditional statement: The conditional statement can be described as a logical statement with the phrase "if-then". For example: "If quadrilateral has four congruent sides and angles, then quadrilateral is a square".

Converse statement: When we flip the order of the original statement, in this case, the converse statement will be created. For example: If a quadrilateral is a square, then quadrilateral will have four congruent sides and angles".

Compound statement: When we add the word 'and' between the above two statements, in this case, the compound statements will be created. For example: "You want to go on a Goa trip and we are here to help".

So simply, we can say that the compound statement, conjunction, or conditional statement with its converse is also the biconditional statement. The bi-conditional statement as a compound statement is described as follows:

"If two line segments are congruent, then the length of these lines segments is equal; and, if length of two line segment is equal, then both the line segments are congruent."

This statement is used to show a true bi-conditional statement because this statement is a combination of conditional: If two line segments are congruent, then the length of these lines segments is equal, with its converse: if the length of two line segments is equal, then both the line segments are congruent.

We can also write the above bi-conditional statement in a simple way, i.e., "The two line segments are congruent if and only if the length of these lines segments is equal".

Examples of Conditional and its Converse:

Example 1:

Conditional: If I scored 65% or more than that, then I passed the exam. (True)

Converse: If I passed the exam, then I scored 65% or more than that. (True)

So we can write two bi-conditional statements because the above conditional statement and the converse statement are true. The two biconditional statements are described as follows:

1. I scored 65% or more than that if and only if I passed the exam.
2. I passed the exam if and only if I scored 65% or more than that.

If the conditional and converse statements have the same truth value, only then we can create two bi-conditional statements. It will not matter whether the truth value is true or false. The concern is only about the same truth value. Both statements could be false, and we can still write a true biconditional statement. ("My pet dog draws an apple photo if and only if my pet dog online purchases the art supplies").

In the next example, we will use our pet dog and see how the logical biconditional can be prevented by the different truth values.

Example 2:

Conditional: If I have a pet dog, then my time to study will be killed. (true)

Converse: If my time to study is killed, then I have a pet dog. (not true)

In the following statements, we attempt to write the logical bi-conditional statement, but we have failed to do this, and these statements also do not make any sense.

• I have a pet dog if and only if my time to study will be killed. (not true)
• My time to study will be killed if and only if I have a pet dog. (not true)

Symbols of Biconditional statements

We may recall that with the help of logic symbols, we are able to replace words in statements. Suppose there is a conditional statement, "If I have a pet dog, then my work to study will be killed". Now we can replace the premise with the letter P, conclusion with the letter Q, and add a symbol → for the connector like this:

We always use the symbol ⇔ for the bi-conditional statements because, in this statement, the truth works in both directions.

The bi-conditional statements are a combination of the two statements, which are described as follows:

This combination can also be represented in another way, which is described as follows:

Truth Table:

When there is a statement, and we have to check whether it is true, we usually prefer to use the truth table, through which we can easily compare the true values (whether these values are true or false). With the help of this table, we can evaluate the logical statement. The biconditional statement "P if and only if Q" will be true if P and Q both have the same truth value in the truth table.

Now we will take our original biconditional statement, i.e.,

"You are reading this article very carefully if and only if you have interest in learning the concept of compound statements, converse statement and truth tables so that it will be easily to know about a true biconditional statement".

From this biconditional statement, the statements of P and Q will be

P: You are reading this article very carefully

Q: You have an interest in learning the concept of compound statements, converse statements, and truth tables so that it will be easy to know about a true biconditional statement

The truth table for the above P and Q statement and the overall statement is described as follows:

Individually, P and Q give the combination of 4 possible truth values because they can either be true or false. The two middle lines of this table is a counterexample of logical biconditional, which says that "You are reading this article very carefully but you does NOT have any interest in learning the concept of compound statements, converse statement and truth tables" and "You did not read this article very carefully but you are interested in learning the concept of compound statements, converse statement and truth tables".

With the help of first and last statement of the above table, the logical biconditional is supported. The first statement is used to show that "You are reading this article very carefully and you are interested in learning the concept of compound statements, converse statement and truth tables". But the last statement does not say anything related to the condition at all, so this statement does not disprove the biconditional. So the fourth statement says that "You are not reading this article very carefully and you are not interested in learning the concept of compound statements, converse statement and truth tables".

Example of Bi-conditional statement

From the above discussion, we have learned several conditional statements and their converse statements.

• Conditional: If I scored 65% or more than that, then I passed the exam. (True)
• Converse: If I passed the exam, then I scored 65% or more than that. (True)
• Conditional: If I am breathing, then I am alive. (True)
• Converse: If I am alive, then I am breathing. (True)

Now we will write the bi-conditional statements for the above described two statements like this:

1. I scored 65% or more than that if and only if I passed the exam
2. I passed the exam if and only if I scored 65% or more than that.
3. I am breathing if and only if I am alive
4. I am alive if and only if I am breathing

We will be able to "clean up" the words for grammar if we will successfully write the converse and bi-conditional for these.

• If I go to Gym, then I will feel relaxed.

Before picking the below, we will try to do it:

• If I go to Gym, then I will feel relaxed. (True)
• If I feel relaxed, then I will go to Gym. (True)

Bi-conditional statements of the above statements are described as follows:

1. I will go to Gym if and only if I feel relaxed.
2. I feel relaxed if and only if I will go to Gym.

Now we will write the other leftover like this:

• If I ask more doubts to my teacher, then I will understand the concept better. (True)
• If I understand the concept better, then I will ask more doubts to my teacher. (False)

The truth values of both the above leftover are not the same, so we cannot write a biconditional statement for this.

Important Points:

There are some important points that we should know when we are learning the biconditional statements. With the help of these points, we can easily identify whether the given statement is a biconditional statement or not. These important points are described as follows:

• When we combine the conditional statement and its converse statement, then a biconditional statement will be created. So one conditional will be true iff the other one is true as well.
• To indicate the biconditional statements, we often use the words "if and only if". Sometimes, we also use its shorthand "iff".
• The biconditional statements always use a double arrow. This arrow is used to show us that the condition must be true in both directions.