Truth Tables Of A Conditional Statement And Its Converse Inverse And Contrapositive
Now that we know how to symbolically write the converse, inverse, and contrapositive of a given conditional statement, it is time to state some interesting facts about these logical statements.
To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table.
Here are some of the important findings regarding the table above:
- The conditional statement is NOT logically equivalent to its converse and inverse.
- The conditional statement is logically equivalent to its contrapositive. Thus, p} \to q} \equiv ~\colorq\to ~\colorp.
- The converse is logically equivalent to the inverse of the original conditional statement. Therefore, q} \to p} \equiv ~\colorp\to ~\colorq.
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What Is Converse Meaning In Maths
In mathematics, the converse of a statement is a statement that is logically equivalent to the original statement, but with the order of the statements reversed.
For example, the statement If it is raining, then the ground is wet is logically equivalent to the statement If the ground is wet, then it is raining.
The converse of a statement is not always true, but it is always logically equivalent to the original statement.
What Are The Converse Contrapositive And Inverse
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Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesnt take long to run into something of the form If P then Q. Conditional statements are indeed important. What is also important are statements that are related to the original conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse.
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What Is Converse In Math
What Is Converse In Math
What Is Converse In Math?
In mathematics, the converse of a statement is a statement that is logically implied by the original statement and its negation. The converse of If A, then B is If not B, then not A.
The converse of a statement is not always true. For example, the converse of If A, then B is If B, then A. However, this statement is not always true. For example, if A is the sky is blue and B is the grass is green, then the statement If the sky is blue, then the grass is green is not always true.
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Truth Value Of A Converse Statement
The truth value of the original statement implies nothing about the truth value of the converse. Likewise, the truth value of the converse implies nothing about the truth value of the original statement.
Indeed, it is possible to have a false statement with a true converse and a false statement with a false converse. Likewise, a true statement can have a false converse or a true converse.
When a statement and its converse are both true, it is called a biconditional statement.
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How Do You Write Converse
Converse is a form of communication in which people exchange ideas, impressions, or feelings. It can be used to discuss a topic, or to simply chat with someone. In order to converse effectively, its important to understand the basics of communication.
Conversation involves two or more people who are engaged in a back-and-forth exchange of ideas. The conversation flows when both parties are actively participating. In order to keep the conversation flowing, its important to listen attentively and respond to the other persons comments.
Conversation can be used to discuss a topic, or to exchange ideas and impressions. It can also be used to build relationships and to make new friends. In order to converse effectively, its important to be aware of your tone of voice, and to be respectful of the other person.
Conversation can be a great way to learn new things, and to get to know people better. Its a social activity that can help to build relationships and to strengthen connections. By being aware of the basics of communication, you can converse more effectively with the people around you.
How Is The Word Converse Used In Math
You may know the word converse for a verb meaning to chat, or for a noun as a particular brand of footwear. Neither of those is how mathematicians use converse. Converse and inverse are connected concepts in making conditional statements. To create the converse of a conditional statement, switch the hypothesis and conclusion.
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Geometry And Conditional Statements
Many times in geometry we see postulates and theorems that seem like they could become conditional statements and converse conditional statements:
- Parallel lines never meet. Postulate
- If two lines are parallel, then they are lines that never meet. Conditional Statement
- If two lines never meet, then they are parallel. Converse
Example #2
- Adjacent angles share a common side. Postulate
- If angles share a common side, then they are adjacent. Conditional Statement
- If angles are adjacent, then they share a common side. Converse
Some postulates are even written as conditional statements:
- If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
- If two points lie in a plane, then the line joining them lies in that plane.
Practice Problems On The Converse Of A Statement
Find the converse of the following statements:
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What Are Contrapositive Statements
It is easy to understand how to form a contrapositive statement when one knows about the inverse statement. To create the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. First, form the inverse statement, then interchange the hypothesis and the conclusion to write the conditional statements contrapositive.
Click here to know how to write the negation of a statement.
In other words, contrapositive statements can be obtained by adding not to both component statements and changing the order for the given conditional statements.
Converse Of A Conditional Statement
The converse of a true conditional statement does not automatically produce another true statement. It might create a true statement, or it could create nonsense:
- If a polygon is a square, then it is also a quadrilateral.
That statement is true. But the converse of that is nonsense:
- If a polygon is a quadrilateral, then it is also a square.
We know it is untrue because plenty of quadrilaterals exist that are not squares.
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Why Do Contrapositive And Converse Proofs Work
Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true. It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. There is an easy explanation for this.
Is the converse and inverse of the conditional statement the same?
It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement If Q then P .
Converse Statements And Beyond
If you use the letters p and q to generalize conditionals statements, here is the quick reference table using p and q to stand in for conditional statements. This is a good reminder of how the contrapositive, inverse, and converse of a statement work.
Once you are familiar with these terms, you will see them used in other math proofs such as the converse of the Pythagorean Theorem, or even in day-to-day life.
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The Inverse Of A Conditional Statement
When youre given a conditional statement p} \to q}, the inverse statement is created by negating both the hypothesis and conclusion of the original conditional statement. Thus, the inverse is the implication ~\colorp\to ~\colorq.
The symbol ~\colorp is read as not p while ~\colorq is read as not q .
Mathematics And Computer Science
The Department of Mathematics and Computer Science, housed in the LEED certified Kuhn Hall, offers students a well-grounded education in quantitative learning and computer skills/programming.
In addition to offering many learning experiences for students who take classes that meet the colleges general education requirements, the department offers majors in mathematics, as well as minors in mathematics and computer science.
All students graduating from Converse must take a mathematics course as part of the General Education Program or can exempt the math requirement by passing an exemption test or by getting a score of 3 or higher on a calculus or statistics AP high school exam.
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When The Original Statement And Converse Are Both True
when two statements are both true or both false. When the converse and inverse of a conditional statement are either both true or both false, then these pairs of statements are called equivalent statements.
How do you know if a converse is true or false?
The lines should never be wobbly. If you have a pair with a line that stops and starts again then this is a problem. The toe cap will be in sturdy rubber and will be completely straight where it meets the lacing. The toe cap on an authentic pair of Converse should not be curved towards the tongue of the shoe at all.
Converse Of A Statement Definition
Let P and Q be the two simple statements, and P Q be the compound statement.
Therefore, the converse of a statement P Q is Q P.
It should be observed that P Q and Q P are converse of each other.
In Geometry, we have come across the situations where P Q is true, and we have to decide if the converse, i.e., Q P, is also true.
Now, let us understand how to find the converse of a statement with the help of an example.
Given statement: If a triangle ABC is an equilateral triangle, then all its interior angles are equal.
To find the converse of a given statement, first we have to identify the statements P and Q.
The given statement is in the form P Q. Now, we have to find Q P.
Here, P = Triangle ABC is an equilateral triangle.
Q = Triangle ABC interior angles are equal.
Hence, the converse of a statement is If all the interior angles of triangle ABC are equal, then it is an equilateral triangle, which is in the form Q P.
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The Converse Of A Conditional Statement
For a given conditional statement p} \to q}, we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. Therefore, the converse is the implication q} \to p}.
Notice, the hypothesis \largep} of the conditional statement becomes the conclusion of the converse. On the other hand, the conclusion of the conditional statement \largep} becomes the hypothesis of the converse.
Exchanging Parts Of Conditional Statements
You can switch the hypothesis and conclusion of a conditional statement. You take the conclusion and make it the beginning, and take the hypothesis and make it the end:
- If my dog observes something that excites him, then he barks.
- If triangles have equal corresponding sides, then they are congruent.
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Converse Inverse And Contrapositive Of A Conditional Statement
What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive.
But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.
Other Types Of Statements
In addition to converses are two other conditional statements derived from a given conditional statement. They are inverses and contrapositives.
The inverse negates both the antecedent and the consequence. The contrapositive is the converse of the inverse. That is it both reverses and negates the antecedent and the consequence of a given conditional statement.
Using logic symbols, for a given statement $P \rightarrow Q$, the inverse is $\neg P \rightarrow \neg Q$, and the contrapositive is $\neg Q \rightarrow \neg P$.
For the converse, inverse, and contrapositive, a conditional reference statement is necessary.
For example, note that the converse of the contrapositive results in the inverse. That is, swapping the antecedent and consequence of the contrapositive results in the inverse.
Note that the contrapositive and the original statement will always have the same truth value. The converse and the inverse will always have the same truth value. On the other hand, the truth values of the original statement and the converse, the original statement and the inverse, the contrapositive and the converse, and the contrapositive and the inverse are independent of each other.
It is also important to note that making any logical statement done twice results in the original statement.
That is, the converse of the converse is the original statement.
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What Is Converse And Inverse
What is converse and inverse?
Converse and inverse are two mathematical terms that are often used together. Converse is a term used in mathematics that refers to a change in a statement that results in the statement being true. Inverse is a term used in mathematics that refers to a change in a statement that results in the statement being false.
What Are Converse Statements
The converse statements are formed by interchanging the hypothesis and conclusion of given conditional statements.
Thus, we can relate the contrapositive, converse and inverse statements in such a way that the contrapositive is the inverse of a converse statement.
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This can be better understood with the help of an example.
Example: Consider the following conditional statement.
If a number is a multiple of 8, then the number is a multiple of 4.
Write the contrapositive and converse of the statement.
Solution:
Given conditional statement is:
If a number is a multiple of 8, then the number is a multiple of 4.
The converse of the above statement is:
If a number is a multiple of 4, then the number is a multiple of 8.
The inverse of the given statement is obtained by taking the negation of components of the statement.
If a number is not a multiple of 8, then the number is not a multiple of 4.
Now, the contrapositive statement is:
If a number is not a multiple of 4, then the number is not a multiple of 8.
All these statements may or may not be true in all the cases. That means, any of these statements could be mathematically incorrect.
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Inverse And Converse Statements
A conditional statement is a statement that has “then” in it. “If it is my birthday, then I will eat ice cream.” Notice there are two things that can be assigned a truth value: It is either your birthday or not, and you will either eat ice cream or you won’t.
In math, this is a true conditional statement. Every time the first condition is true , the second condition is also true. But what is the converse of the statement and what is its inverse?
The converse of the statement is when you change the order of the first statement. “If it is my birthday, then I will eat ice cream” becomes “If I eat ice cream, then it is my birthday.”
The inverse of the statement is when you apply a negation to both statements. “If it is my birthday, then I will eat ice cream” becomes “If it is not my birthday, then I will not eat ice cream.”
What Is A Converse Statement
A converse statement is a conditional statement in which the antecedent and consequence of a given conditional statement are reversed.
Recall that a conditional statement is one that is in the format if, then They can be true or false.
The antecedent of a conditional statement is the part that follows the word if. Similarly, the consequence is the part that follows the word then.
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Converse Of A Statement
Generally, a compound statement is a combination of two or more simple statements. The simple statements can be converted into compound statements using multiple ways. In Mathematics, we have seen many theorem statements which are connected using the words if and then. Now, let us discuss how to find the converse of a compound statement that uses the words if and then with the help of examples.
Converse Of A Theorem
In mathematics, the converse of a theorem of the form PQ will be QP. The converse may or may not be true, and even if true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse was proved only in 1997.
In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of “Given P, if Q then R“ will be “Given P, if R then Q“. For example, the Pythagorean theorem can be stated as:
Given a triangle with sides of length a
In traditional logic, the process of switching the subject term with the predicate term is called conversion. For example going from “No S are P” to its converse “No P are S”. In the words of Asa Mahan:
“The original proposition is called the exposita when converted, it is denominated the converse. Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita.”
The “exposita” is more usually called the “convertend.” In its simple form, conversion is valid only for E and I propositions:
Type | ||
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Some S is not P | not valid |
The validity of simple conversion only for E and I propositions can be expressed by the restriction that “No term must be distributed in the converse which is not distributed in the convertend.” For E propositions, both subject and predicate are distributed, while for I propositions, neither is.
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