2. - Conditional statements are used in every field
of human endeavor.
- They are crucial to the search for truth
- If you are going to be able to adequately
interpret and judge the statements you hear,
you must understand the structure of
Conditional statements.
-
WARNING: Once you get good at these, you
may be amused by statements made by
politicians and other public speakers.
3. Objectives
Students will be able to:
Identify conditional statements and place them in
if-then form
Identify the hypothesis and conclusion of a
conditional statement
Convert conditional statements into their other
logical variants
Identify and use truth relationships of conditional
statements
State and use the point, line and plane postulates
of geometry
4. What is a Conditional
Statement?
A statement that can be written in the
format:
If …., then ….
The part after “if” is called the hypothesis
Do not confuse this with the word “hypothesis”
from science
The rest (after “then”) is the conclusion
5. Statements Not in If-Then Form
- Often statements are not in if –then form, but to
test them out scientifically, we must convert them
- In English, there are infinite ways to rephrase a
conditional statement; however, we will cover the
three most common variations here
6. Standard Sentence
Split the subject and predicate.
Add “If it is” or “If they are” to the subject
Add “then it” or “then they” to the predicate
Smooth out the grammar
Example: “All dogs go to heaven.”
“If they are dogs, then they go to heaven.”
7. “Whenever” or “When”
Replace “whenever” or “when” with “if”
Add “then” after the comma
Example: “Whenever I see a seagull, I think of
home.”
If I see a seagull, then I think of home.
8. “If” at the end
Move the “if” clause to the beginning
Add “then” after the “if” clause
Example: “I eat if I am hungry.”
“If I am hungry, then I eat.”
9. Logical Variations of the
Conditional
-Often a conditional statement can be difficult to
prove or unwieldy to use.
- By using logical variations, we find forms easier to
prove or use.
10. Converse, Inverse, &
Contrapositive
Converse: formed by swapping the
hypothesis and the conclusion
Inverse: formed by negating the hypothesis
and conclusion
Contrapositive: formed by both negating and
swapping the hypothesis and conclusion
11. Equivalent Statements
If the Conditional is true (or false) then so is
the Contrapositive and vice versa.
Similarly, if the Converse is true, then so is the
Inverse and vice versa
If both the Conditional and its Converse are
true, then they can be rewritten as a
Biconditional statement (more next class)
12. Example 1
If you added 2+2, you got 4
CONVERSE:
If you got 4, then you added 2+2.
INVERSE:
If you did not add 2+2, you did not get 4.
CONTRAPOSITIVE:
If you did not get 4, then you did not add 2+2.
13. Example 2 (the word “not”)
If you do not eat, you will be hungry
CONVERSE:
If you are hungry, then you did not eat.
INVERSE:
If you ate, then you are not hungry.
CONTRAPOSITIVE:
If you are not hungry, then you ate.
15. Point, Line & Plane Postulates
5. Through any two points there exists exactly one line.
6. A line contains at least two points.
7. If two lines intersect, then their intersection is exactly
one point.
8. Through any three noncollinear points there exists
exactly one plane.
9. A plane contains at least three noncollinear points.
10. If two points lie in a plane, then the line containing
them lies in the plane.
11. If two planes intersect, then their intersection is a
line.
-Pay attention to how the word “not” acts in this statement- Often you must adjust the tense of the verb or add helping verbs to make the sentence “sound” right without changing the logical structure of the sentence