1. Logical Sequencing & Conditional Statements

2. Conditional Statements
A conditional statement is written in the form if p, then q.
If a given condition is met (if p), then another condition is true or an event will happen (then q).
The if-clause is the hypothesis; the then-clause is the conclusion.

3. Conditional Statements
Ex 1.) If you don’t do your homework, then
you will get a zero.
Ex 2) Rewrite the statement in if-then form:
2.) Every multiple of 4 is also a multiple of 2.
If a number is a multiple of 4, then it is a multiple of 2.
True or False?
True
p - Hypothesis
q - Conclusion

4. Logical Order
When given several related conditional statements, you must put them in logical order. The conclusion of one statement will flow into the hypothesis of the next.

5. Logical Order – Put the following if-then statements in order
A. If Cameron graduates with a degree, then he will make a lot of money.
B. If Cameron studies hard, then his grades will be good.
C. If Cameron makes a lot of money, then he will be able to buy a new car.
D. If Cameron attends college, then he will graduate with a degree.
E. If Cameron has good grades, then he will be able to attend college.
Logical Order:
Conclusion: If Cameron studies hard, then he will be able to buy a new car.

6. Logical Order Ex. 2 – Put the statements in logical order.
A. If a shape is a square, then it is a rhombus.
B. If a shape is a parallelogram , then it is a quadrilateral.
C. If a shape is a quadrilateral, then it is a polygon.
D. If a shape is a rhombus, then it is a parallelogram.
Conclusion: If a shape is a square, then it is a polygon.

7. Converse
What is the converse?
The converse of a conditional statement is formed by switching the places of the hypothesis and conclusion.
The sentence if p, then q
becomes if q, then p.
Is the given statement true or false? Write the converse statement for each conditional statement.
Is the converse true or false? If the converse is false, come up with a counter example.

8. Examples:
1. If a quadrilateral is a rectangle, then it is a parallelogram.
TRUE
If a quadrilateral is a parallelogram, then it is a rectangle. False
Counterexample: It could be a rhombus

9. Inverse Ex: If it is sunny outside, then I will go running.
The inverse of a conditional statement is formed by negating the hypothesis and the conclusion.
The sentence if p, then q becomes
if not p, then not q.
What is the Inverse?
Ex: If it is sunny outside, then I will go running.
Becomes… If it is not sunny outside, then I will not go running.

10. 4. Example: If two lines are perpendicular, then they intersect.
Is the given statement true or false? Write the inverse statement for each conditional statement. Is the inverse true or false? If the inverse is false, come up with a counter example.
4. Example: If two lines are perpendicular, then they intersect.
TRUE
If two lines are not perpendicular, then they don’t intersect.
False
Counterexample: Two lines that intersect at a 30 degree angle
(They are not perpendicular and they still intersect.)

11. Contrapositive
What is a contrapositive Statement
The contrapositive of a conditional statement is formed in two steps.
1. Form the converse of the statement
2. Form the inverse of the converse.
In other words, the sentence if p, then q becomes if not q, then not p.
Is the given statement true or false? Write the contrapositive statement for each conditional statement.
Is the contrapositive true or false? If the contrapositive is false, come up with a counter example.

12. Examples of Contrapositives
7. If a polygon has just four sides, then it is a quadrilateral.
If it is not a quadrilateral, then it is not a polygon with four sides.
8. If 2 angles form a straight line, then their sum is
If the sum is not , then the 2 angles don’t form a straight line.