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Table of contents
1.
Introduction
2.
Conditional Statements
2.1.
Case 1 (True implies True)
2.2.
Case 2 (True implies False)
2.3.
Case 3 (False implies both True and False)
3.
Converse Statement
4.
Inverse Statement
5.
Contrapositive Statement
6.
Bi-Conditional Operation
7.
Principle of Duality
8.
FAQs
9.
Key Takeaways
Last Updated: Mar 27, 2024

Conditional and Biconditional statements

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Ashwin Goyal
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Introduction

In this blog, we will discuss Conditional and biconditional statements and discuss some terms related to them. But before we discuss this topic in detail, we should first look at some of the basic terminologies related to this topic.

Compound Statement: A compound statement is formed by combining two basic assertions with conditional terms such as 'and,' 'or,' 'not,' 'if,' 'then,' and 'if and only if.'

Compound Proposition: A compound proposition is a statement made up of two propositions connected by the connective "If...then...". p→q is denoted.

Conditional Statements

Conditional Operation occurs when a compound statement is generated by two basic assertions linked by the phrase 'if and then.' A conditional statement can be broken down into two statements: a hypothesis, and the other is a conclusion statement. For example, consider the following statement: "If you practice DSA on Coding Ninjas Studio, then you clear any interview of FAANG." This statement is made up of two simple statements:

 p:" If you practice DSA on Coding Ninjas Studio."

 q: "then you can clear any interview of FAANG."

Here p is called a hypothesis, and q is a conclusion statement.

Remember that hypothesis always starts with "if," and conclusion always starts with "then."

According to the original statement, if p is true, then only q is true, or we can say in layman terms that if then q. This can also be rephrased as p implies q, p→q.

The truth table for Conditional Operation:

P

Q

 Q

T

T

T

T

F

F

F

T

T

F

F

T

 

 

Now let us discuss some example statements to get a better idea of conditional statements. 

Case 1 (True implies True)

If p and q both are true, then p→q is true. 

For instance: If 1+10 = 11, then the moon will revolve around the earth.

Here p is "If 1+10 = 11" and q is" then the moon will revolve around the earth." As we can see here, p is true, and q is also true; therefore, p→q is also true.  

Case 2 (True implies False)

If p is true and q is false, then p→q is false. 

For instance: If it rains, then I will carry an umbrella.

Here p is" If it rains" and q is "then I will carry an umbrella." In other words, we can rephrase this statement as to when it is raining, and then I will be carrying an umbrella. Now there might be someday when it is raining (true), and I would forget to carry an umbrella (false), then this condition of p→q would be false. 

Case 3 (False implies both True and False)

Whether q is false or not, If p is false, then p→q is true. 

For instance: If chocolate is made of gold, then Neil Armstrong was the first person to step on the moon.

Here p is "If chocolate is made of gold" and q is "then Neil Armstrong was the first person to step on the moon." As we can see, p is false here, but q is true. Therefore, statement p→q is true.

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Converse Statement

The statement q→p is referred to as the converse of the statement p→q. A converse statement and a conditional statement are not the same things.

Inverse Statement

The statement ~p→~q is referred to as the inverse of the statement p→q. An inverse statement and a conditional statement are not the same things.

Contrapositive Statement

The statement ~q→~p is referred to as the contrapositive of the statement p→q.

To better understand the above statements, let us look at a sample example.

Consider the following conditional statement:
           "If today is Sunday, then yesterday was Saturday."
Here p is "If today is Sunday," and q is "then yesterday was Saturday."
 
Now we can create the above-discussed statements:
Converse Statement: "If yesterday was Saturday, then today is Sunday."
Inverse Statement: "If today was not Sunday, then yesterday was not Saturday."
Contrapositive Statement:" If yesterday was not Saturday, then today is not Sunday."

 

Have a look at this sample question to understand the concept of conditional statements.

Q) Prove that commutative law does not hold for conditional statements. In other words, prove that p→q is not equivalent to q→p.

P

Q

 Q

 P

T

T

T

T

T

F

F

T

F

T

T

F

F

F

T

T

Here the last two columns from the right are different, i.e., the column of p→q is not the same as the column for q→p; therefore, we can say that p→q is not equivalent to q→p. Hence proved that commutative law is not true for conditional statements.

Bi-Conditional Operation

Bi-Conditional Operation is represented by the symbol "↔." Bi-conditional Operation occurs when a compound statement is generated by two basic assertions linked by the phrase 'if and only if.'  Biconditional statements are true only if both p and q are true or false.

The truth table for Bi-conditional Operation:

P

Q

↔ Q

T

T

T

T

F

F

F

T

F

F

F

T

 

For example: 

"A rectangle is a square if and only if all the sides are equal."

Principle of Duality

According to this theorem, the dual of a Boolean function is derived by replacing the logical AND operator with the logical OR operator and zeros with ones. There will be a Dual matching function for each Boolean function.

FAQs

  1. What is a mathematical statement?
    It is the fundamental unit of all mathematical reasoning. Furthermore, mathematical reasoning might be inductive (mathematical induction) or deductive (mathematical deduction). Any forceful language that may be claimed to be true or untrue but not both indicates that it is a mathematically admissible assertion. As a result, this form of statement is correct.
     
  2. What is the equivalence of propositions?
    Equivalence of propositions is said when two propositions have the same values under all the conditions.     
     
  3. What is satisfiability?
    A compound proposition is satisfiable if at least one TRUE result is in the truth table.

Key Takeaways

This article briefly discussed Conditional and Biconditional statements, and we have also discussed some different types of statements under conditional statements.

I hope you must have gained some insight into this topic of Conditional and Biconditional Statements, and by now, you must have developed a clear understanding of them. You can learn more about such topics on our platform Coding Ninjas Studio.

Thank you for reading.

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Basic Logical Operations
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Binary Relations
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