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).
So, the answer is
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)
So, the answer is,
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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