Contrapositive and converse of a statement

Introduction

The word contrapositive and converse can be used and applied only to implicated statements that are statements of the type “if p, then q.” The converse and contrapositive statements are statements in their very own rights. However, we used the words converse and contrapositive to refer to the converse and contrapositive of something that has already been declared a statement.

If p implies q is a statement, its converse is ”q implies p”. So basically, to create the converse of an argument, all there is to do is reverse the order of the sentences and keep the implication between the two.

Statement – If p, then q

Converse – If q, then p

Inverse – If not p, then not q

Contrapositive – If not q, then not p 

For example: If it is raining, I carry an umbrella. 

Converse: if I have an umbrella, it is raining.

Observation: If p implies q is true, it does not necessarily mean that q suggests p is also true, i.e., if a statement is true, its converse notice may or may not be accurate. For example, every man is a human. So I converted the converse of an argument if p implies q is that if he is a man, he is a human.

Converse: If he is human, he is a man who can be written in the general form as every human is a man. With their common sense, this statement is false. However, its public information is accurate, such as every man is a human. It states that if p implies q is true, it doesn’t mean that its converse q implies p is also true.

Transitivity in implications

When you have a relationship between two statements given in a specific order, you can generate a relation out of any two of those, i.e., out of any two of the hypotheses or the conclusions, providing you maintain the proper order. It can mislead at times, as we saw after some time. Now the transitivity which applies is:

If P implies q and then q, which is the conclusion of the first statement, implies r, then p implies r, i.e., if a hypothesis suggests a conclusion and conclusion applies some other conclusion, then the first hypothesis suggests the last decision. 

For example: 

”x is divisible by four implies ”x is divisible by 2 (q implies r), x is divisible by eight means, x is divisible by 2. (p implies r).

When you applied transitivity to what you have been given about an implicated statement in the correct order, then certain conclusions arrived at a specific hypothesis.

General misconceptions:

You have to avoid these misconceptions to make valid statements in mathematics:

• Something which arises out of a flawed understanding of what transitivity is if you applied transitivity and what you have been given about the implicated statement in a wrong way or in a way that is flawed in the order of the hypothesis and conclusion, you are arriving at misleading results as we were seen with an example
• However, the general misconception is that if p implies q and p imply r, it does not mean that q implies r. 

To understand this, we are taking elementary examples quite much nothing to do with mathematics, but it helps to identify how deep the flaw can be:

For example: if it is a parrot, then it is green. (This is true)

If it is a parrot, then it is a bird. (This is true)

But now if I take the conclusions of both and make an implicated statement what it turns out to be:

If it is green, then it is a bird. (False statement) so, if the same arguments conclude two arguments, they suppose they do not imply each other.

CONTRAPOSITIVE: 

The contrapositive is a term that refers to an implicated statement. Contrapositives are statements in their rights. Now the contrapositive of p implies q is not p means not q. So here, we reverse the information order and take the negations (“not p” is the same as ” negation of p”).

Contrapositive statement example: Every crow is black.

Converse: If it is a crow, it is black.

Contrapositive: If it is not black, it is not a crow.

i.e., everything that is not black is not a crow. 

Observations:

This fact will apply to every contrapositive statement. The contrapositive of an account is logically equivalent to the original idea. Equivalence means that given a piece of information, any constructive contrapositive. So, both the words and their contrapositive imply the same meaning.

Logical Equivalence 

If you see the examples stated in the above paragraphs, you’ll notice the pattern of the conditional statements and how they are conditioned to the initial reports. It proves that the conditional statement has a truth to its contrapositive. Therefore, these statements are equivalent logically. However, the conditional checks are not logically equal to the inverse and its converse. 

The conditional statements and the contrapositive are logically equivalent, and it is an advantage when proving the theorems in mathematics. Rather than proving the theorem directly, we can indirectly prove the conditional statement by proving the contrapositive of the information. It is easier and more effective. 

Conclusion 

To conclude, we can say that the word contrapositive and converse can be used and applied only to implicated statements that are statements of the type “if p……. then q”. If p implies q is a statement, its converse is ”q implies p”. So basically, to create the converse of an argument, all there is to do is reverse the order of the sentences and keep the implication between the two.