Converse Statement

A converse statement is a new conditional statement which is formed by reversing the hypothesis and conclusion of the original “if–then” statement, meaning the order is flipped from P → Q to Q → P.

For example,
Original statement: “If a number is even, then it is divisible by 2.”
Converse statement: “If a number is divisible by 2, then it is even.”

Using symbols:
If a statement is P → Q, then its converse is Q → P.
 

Follow these steps to write a converse statement:

Step 1: Identify hypothesis and conclusion

Step 2: Swap them

Step 3: Write in conditional form

A few examples of true converse statements are:

Original statement: “If it is morning, then the sun has risen”

Converse statement: “If the sun has risen, then it is morning” 

Original: “If a triangle is equilateral, then all three sides are of the same length.”

Converse: “If all three sides are of the same length, then the triangle is equilateral.”

This converse is also true by definition of an equilateral triangle.

Some false converse statements are: 

Original statement: “If it’s a cat, then it must be a mammal.”

Converse statement: “If it is a mammal, then it must be a cat.” (This statement is false because other animals like dogs and elephants are also mammals).

Original statement: “If it is a rectangle, then it should have four sides.”

Converse statement: “If a shape has four sides, then it is a rectangle.” (this statement is false because a square also has four sides).

Inverse statements are the inverse of a conditional statement. It is formed by negating both the hypothesis and the conclusion of the original statements. 
 

A contrapositive statement is formed by swapping the conclusion and hypothesis of a conditional statement and then negating both.
 

To understand converse, inverse, and contrapositive statements, we start with the original conditional statement:
 

Original Statement:
 

If p, then q → p → q
Here, p is the hypothesis and q is the conclusion.
 

From this, we form:
 

Converse: q → p
 

Inverse: ¬p → ¬q
 

Contrapositive: ¬q → ¬p
 

Below are the truth tables in a simple and combined format:
 

Converse statements can be a difficult topic to get a grasp on and this section we will discuss some tips and tricks that can help us master this topic.
 

• Before writing the converse, make sure you clearly understand the original conditional statement — the “if–then” structure.

• The converse is formed by reversing the hypothesis and conclusion.

• Remember that a converse is not always true even if the original statement is.

• Practice with everyday scenarios — like sports, weather, or school situations.

• To truly master converse statements, learn how they differ from inverse and contrapositive forms.
   

• Help students to understand that the truth of a conditional does not guarantee the truth of its converse. Use examples where the converse is false to develop reasoning.
   

• Explain that a converse is made by simply switching P and Q in a statement.
   

• Use everyday situations to show how a converse works. For instance, if it rains, the ground gets wet. In Converse: If the ground is moist, it must have rained. (Not always true). This helps children understand that converse statements are not always accurate, even if the original is.
   

• Remind students that writing a converse is like rewriting the sentence with the order changed.
  This helps students who are strong in language but weak in logic.

Converse statements are used for logical reasoning. Here are a few real-world applications of where we use converse statements:

• Education: Many educational institutions might use converse statements for certain policies. In situations such as passing criteria or attendance.

• Insurance: Insurance companies also use converse statements for situations where they would want to check whether their customers pass certain criteria to claim a discount on the insurance.

• Sports: Coaches and sports analysts use converse to assess performance. For example, “if a referee is skilled, then he makes the right decision 80% of the time.” For this, the converse would be, “if a referee makes the right decision 80% of the time, then he is skilled.” This will help an analyst evaluate whether consistent performance indicates skill.

• Technology: If a website loads slowly, then the server is busy. The converse statement for this would be, "If the server is busy, then the website loads slowly." 
   
• Healthcare: In medicine, doctors often use converse reasoning to test hypotheses.