Converse, Inverse, Contrapositive




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Converse, Inverse, Contrapositive


Establish that if a statement is true, the contrapositive is also logically true. Likewise, when the converse is true, the inverse is also logically true.

We start with a simple statement of fact, like, (1) A triangle is a polygon, or (2) An even number is divisible by two. Each of these is actually an implication, an if-then statement: (1) If an object is a triangle then it is a polygon. (2) If a number is even then it is divisible by two. Here is a quick definition of "converse," "inverse," and "contrapositive:"



  • statement: if p then q

  • converse: if q then p

  • inverse: if not p then not q

  • contrapositive: if not q then not p

I will now show the converse, inverse, and contrapositive of our examples involving triangles and even numbers:

Converse: (1) If an object is a polygon then it is a triangle (false). A square is a polygon but not a triangle. (2) If a number is divisible by two then it is even (true). Of course the first one is false because not all polygons are triangles.

Inverse: (1) If an object is not a triangle then it is not a polygon (false). A square is not a triangle, but is a polygon. (2) If a number is not even, then it is not divisible by two (true).

Contrapositive: (1) If an object is not a polygon, then it is not a triangle (true). (2) If a number is not divisible by two then it is not even (true). The first one, being true, cannot be equivalent to the converse, which is false.

Source: http://www.jimloy.com/logic/converse.htm

Second source:

Given an if-then statement "if p, then q", we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause.  For instance, “If it rains, then they will cancel school.” 
              “It rains, “is the hypothesis.
              “They will cancel school,” is the conclusion.

To form the converse of the conditional statement, interchange the hypothesis and the conclusion.


            The converse of “If it rains, then they will cancel school” is “If they cancel school, then it rains.”

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.


            The inverse of “If it rains, then they will cancel school” is “If it does not rain, then they do not cancel school.”

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. 


            The contrapositive of “If it rains, then they will cancel school” is “If they do not cancel school, then it does not rain.”

Statement

If p, then q.

Converse

If q, then p.

Inverse

If not p, then not q.

Contrapositive

If not q, then not p.

 

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.



Example:

Statement

If a person is 18 years old, then he is a legal adult.

Converse

If a person is a legal adult, then he is 18 years old.

Inverse

If a person is not 18 years old, then he is not a legal adult.

Contrapositive

If a person is not a legal adult, then he is not 18 years old.

 

Source: http://hotmath.com/hotmath_help/topics/converse-inverse-contrapositive.html

Find the converse, inverse, and the contrapositive for each of the following:

1) If you build it, they will come.



2) If a city is home to a university then it is great.
3) Taking exams leads to anxiety.
4) Rational numbers are not irrational.
5) Those people who are artistic do well in mathematics.


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