1. Discrete Mathematics(3406)
Inverses converses and Contrapositives

2. Recall: Conditional Statement IF ________________
There are two other types of conditionals called the inverse and the contrapositive.
Recall: Conditional Statement IF ________________
Converse of Conditional IF ________________
Contrapositive IF _________________________
Inverse IF _________________________P (if)Q (then)
Adding “not” to a statement is called ______________ the statement.
Here are some other examples with their negation. p
not q
is parallel
is even =
not p

3. Venn diagrams can represent conditionals and can be very helpful in
understanding conditional statements.
If an item is in circle p it is also in circle q. if p then q (conditional)
Also, if an item is not in circle q then it is not in circle p. if not q then not p (contrapositive)
Thus, the conditional and the contrapositive are said to be ___________________ statements.
If the conditional is true, then the contrapositive is ________.
Since they are the same, you can prove a conditional by proving its contrapositive!
Sometimes this is actually easier than proving the conditional itself.

4. Conditional Contrapositive Converse Inverse
The Venn diagram on the right represents the converse and the inverse.
If an item is in circle q then it is in circle p. if q then p (converse)
If an item is not in circle p then it is not in circle q. if not p then not q (inverse)
For this reason, the converse and inverse are ___________________ statements.
If the converse is true, then the inverse is _________.
To summarize all possible relationships between conditionals:
Conditional
Contrapositive
Converse
Inverse
Sometimes we have to re-word a statement from standard English to fit If-Then form.
All squares are rhombuses  If ____________________________________
All marathoners have stamina  If ______________________________________

5. Sample problems:
Write the contrapositive, converse and inverse of each conditional.
If 2x + 1 = 7, then x = 3.
Contrapositive:________________________________________________________
Converse:____________________________________________________________
Inverse:_____________________________________________________________
If a polygon has five sides, then it is a pentagon

6. Classify each conditional as true or false
Classify each conditional as true or false. Give its converse, inverse and contrapositive.
Then state if each is T/F?
If a triangle is isosceles, then it is equilateral.
Conv______________________________________________________
Inv_______________________________________________________
Contra____________________________________________________
If mA  90, then A is not a right angle.
If today is February 29, then tomorrow is March 1

7. Assume the given statement is true
Assume the given statement is true.What can you conclude by using the given statement
together which each additional statement? If no conclusion is possible, say so.
\If WXYZ is a rhombus, All poets are philosophers.
then its diagonals are perpendicular a.
a. Jose is a poet
b. WXYZ is a square b. Jane is a philosopher
c. mXWY + mWXZ = c. Jung is not a poet.
d. WXYZ is not a rhombus d. Jean is not a philosopher. 

8. Reviewing Conditional Statements
Your Dad says, “If you get a B average, then you can get your driver’s license.”
This is an example of an if-then statement, which is also called a conditional.
You have already learned about the converse of a conditional. It is formed by interchanging the hypothesis and the conclusion.

9. Converse Statement: If p, then q. Converse: If q, then p.
The converse of a conditional is formed by switching the hypothesis and the conclusion:
Statement: If p, then q.
Converse: If q, then p.
hypothesis conclusion

10. Today you will learn about other related conditionals…….
the inverse
the contrapositive

11. GIVEN STATEMENT: If p, then q.
INVERSE: If not p, then not q.
Inverse negates given statement.
CONVERSE: If q, then p.
CONTRAPOSITIVE: If not q, then not p.
Contrapositive negates converse.

12. GIVEN STATEMENT: If p, then q.
If today is Tuesday, then tomorrow is Wednesday.
INVERSE: If not p, then not q.
If today is not Tuesday, then tomorrow is not Wednesday.
CONVERSE: If q, then p.
If tomorrow is Wednesday, then today is Tuesday.
CONTRAPOSITIVE: If not q, then not p.
If tomorrow is not Wednesday, then today is not Tuesday.

13. P A statement and its contrapositive are Q
logically equivalent (either both true or both false.) Q P
Statement: If p, then q.
Contrapositive: If not q, then not p.

14. Q The converse and the inverse are also P
logically equivalent (either both true or both false.) P Q
Converse: If q, then p.
Inverse: If not p, then not q.

15. A statement is NOT
logically equivalent to its converse or to its inverse. Q P
Statement: If p, then q.
True
Converse: If q, then p.
Not true!
Inverse: If not p, then not q.
Not true!

16. What can you conclude about each additional statement?
Example
Suppose this conditional is true:
All runners are athletes.
(If a person is a runner, then that person is an athlete.)
What can you conclude about each additional statement?
Steven is a runner.
Sally is not an athlete.
Susan is an athlete.
Stan is not a runner.
Venn diagrams can be useful to illustrate.

17. All runners are athletes.
Sally
Statement:
All runners are athletes.
Athletes ?
Runners ?
Steven is a runner.
Steven
So, Steven is an athlete.
Sally is not an athlete. ?
So, Sally is not a runner.
Susan
Susan is an athlete. ?
Stan
No conclusion follows.
Stan is not a runner.
No conclusion follows.

18. Classify the statement as true or false
Classify the statement as true or false. Then give the following, and classify each as true or false: (a) converse (switch p and q ) (b) inverse (negate the statement) (c) contrapositive (negate the converse)
If two lines are parallel, then they do not intersect.
True
(a) converse: If two lines do not intersect, then they are parallel.
False
(b) inverse: If two lines are not parallel, then they intersect.
False
(c) contrapositive: If two lines intersect, then they are not parallel.
True

19. Classify the statement as true or false
Classify the statement as true or false. Then give the following, and classify each as true or false: (a) converse (switch p and q ) (b) inverse (negate the statement) (c) contrapositive (negate the converse)
If two angles are acute, then they are complementary.
False
(a) converse: If two angles are complementary, then they are acute.
True
(b) inverse: If two angles are not acute, then they are not complementary.
True
(c) contrapositive: If two angles are not complementary, then they are not acute.
False