# CONVERSE INVERSE AND CONTRAPOSTIVE

Generally the conditional if p then q is the connective most often used in reasoning.

However; with some changes in words in the original statement, additional conditionals can be formed. These new conditionals are called the inverse, the converse, and the contrapositive.

Definition of inverse :

Inverse is a statement formed by negating the hypothesis and conclusion of the original conditional. Symbolically, the inverse is written as (~p ⇒ ~q)

Example :

Right angle is defined as- an angle whose measure is 90 degrees. If you are to write it as inverse statement, it can be done like: If an angle is not a right angle, then it does not measure 90.

Definition of converse :

Converse is a statement formed by interchanging the hypothesis and the conclusion i.e. original conditional (p ⇒ q) is written as (q  p)

Example :

"If two lines don't intersect, then they are parallel", it can be written as "If two lines are parallel, then they don't intersect."

Note : a conditional (p  q) and its converse (q ⇒ p) may or may not be true. It is important that the truth value of each converse is judged on its own merits.

Definition of contrapositive :

Contrapositive is a statement formed by negating both the hypothesis and conclusion (p q) and also then interchanging these negations (~ q ⇒ ~p).

Example :

the statement ‘A triangle is a threesided polygon’ is true; its contrapositive, ‘A polygon with greater or less than three sides is not a triangle’ is true too.

Note :

Remember: a conditional (p ⇒ q) and its contrapositive (~ q ⇒ ~p) must have the same truth value. When a conditional is true, it's contrapositive is also true and when a conditional is false, it's contrapositive is also false.

Problem 1 :

What is the inverse of the statement “If two triangles are not similar, their corresponding angles are not congruent”?

Solution :

Considering the original statements as p and q, its inverse statement can be written in the form (~p ⇒ ~q).

So, the inverse statement is,

If two triangles are similar, their corresponding angles are congruent.

Problem 2 :

What is the inverse of the statement “If it is sunny, I will play baseball”?

Solution :

Considering the original statements as p and q, its inverse statement can be written in the form (~p ⇒ ~q).

If it is not sunny, I will not play baseball.

Problem 3 :

What is the inverse of the statement “If Mike did his homework, then he will pass this test”?

Solution :

If Mike did not do his homework, then he will not pass this test.

Problem 4 :

What is the inverse of the statement “If Julie works hard, then she succeeds”?

Solution :

If Julie does not work hard, then she does not succeed.

Problem 5 :

What is the inverse of the statement “If I do not buy a ticket, then I do not go to the concert”?

Solution :

If I buy a ticket, then I go to the concert.

Problem 6 :

Which statement is the inverse of "If the waves are small, I do not go surfing"?

Solution :

If the waves are not small, I go surfing.

Problem 7 :

Which statement is the inverse of “If x + 3 = 7, then x = 4”?

1) If x = 4, then x + 3 = 7.      2) If x ≠ 4, then x + 3  7.

3) If x + 3  7, then x  4.        4) If x + 3 = 7, then x  4.

Solution :

If x + 3  7, then x  4 is the inverse of the given statement.

Problem 8 :

What is the converse of the statement “If it is sunny, I will go swimming”?

Solution :

Considering the original statements as p and q, (p ⇒ q) its converse statement is written as (q  p)

If I go swimming, it is sunny.

Problem 9 :

Which statement is the converse of “If it is a 300 ZX, then it is a car”?

Solution :

Considering the original statements as p and q, (p ⇒ q) its converse statement is written as (q  p)

If it is a car, then it is a 300 ZX

Problem 10 :

What is the converse of the statement "If it is Sunday, then I do not go to school"?

Solution :

If I do not go to school, then it is Sunday.

Problem 11 :

What is the converse of the statement "If Alicia goes to Albany, then Ben goes to Buffalo"?

Solution :

If Ben goes to Buffalo, then Alicia goes to Albany.

Problem 12 :

What is the converse of the statement "If the Sun rises in the east, then it sets in the west"?

Solution :

If the Sun sets in the west, then it rises in the east.

Problem 13 :

What is the converse of the statement "If x is an even integer, then (x + 1) is an odd integer"?

Solution :

If (x + 1) is an odd integer, then x is an even integer.

Problem 14 :

What is the contrapositive of the statement, “If I am tall, then I will bump my head”?

Solution :

Considering the given statements as p and q, its contrapositive statement is written in the form (~ q ⇒ ~p).

If I do not bump my head, then I am not tall.

Problem 15 :

What is the contrapositive of the statement “If I study, then I pass the test”?

Solution :

Considering the given statements as p and q, its contrapositive statement is written in the form (~ q ⇒ ~p).

If I do not pass the test, then I do not study

Problem 16 :

Given the statement, "If a number has exactly two factors, it is a prime number," what is the contrapositive of this statement?

Solution :

If a number is not a prime number, then it does not have exactly two factors.

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