1. The Conditional And Circuits
Section 3-3
Book Page 117

2. The Conditional and Circuits
Today, we will discuss
Conditionals
Negation of a Conditional
Circuits

3. Conditional Statement…
a compound statement that uses the connective
if…then.
It is written with an arrow p → q
We read the above “p implies q”
or “if p then q.”
The proposition p is the antecedent,
The proposition q is the consequent.

4. Conditional Statements from Geometry
A conditional statement is a statement that can be written in if-then form.
Hypothesis: the phrase following the word if (antecedent)
Conclusion: the phrase following the word then (consequent)
Notation: p  q
(read “p implies q”)
If it rains, then the grass is wet.
Hypothesis: it rains
Conclusion: the grass is wet

5. Not all conditional statements are written using if-then form.
Can Be Expressed…
1. If it is a doggy, it is fuzzy
2. It is fuzzy, if it is a doggy.
3. The doggy is fuzzy.
Given the following statements:
P: The pet is a doggy
Q: It is fuzzy
Not all conditional statements are written using if-then form.
To rewrite the statement in if-then form, you need to identify the antecedent and consequent.

6. Rewriting Conditional Statements
WARNING: The rewrite of the statement in if-then form is not always a word-for-word translation of the original statement.
“All poodles love haircuts.”
can be rewritten as
“If a dog is a poodle, then the dog loves haircuts.”
Give me the rectangular prism look please.

7. Try These…
You make bath- time lots of fun, if you are the rubber ducky.
Aristeetle isn’t really Greek.
No Sasquatches live in Ocala.
Heagy snow skis every Christmas break.
If you are a rubber ducky, then you make bath-time lots of fun.
If the person is Aristeetle, then the person isn’t really Greek.
If the individual is a Sasquatch, then the individual does not live in Ocala.
If it is Christmas break, then Heagy will be snow skiing.

8. Consider the following…
p q
p→q
T T T
T F F
F T
F F
If bubba earns an A, then his parents will give him money.
Possibility
Earned an A?
Received Money? 1 2 3 4
Yes
Yes p is T, q is T
Yes
No p is T, q is F
Yes
Yes p is F, q is T No
No p is F, q is F

9. Truth Table for The Conditional, If p, then q
p q
T T
T F
F T
F F T F T T

10. Emphasize the Following…
Conditional connectives do not imply cause and effect relationships!
Any two propositions may be connected conditionally.
Example:
If it is Saturday,
then football is on TV.

11. Special Characteristics of Conditional Statements
p → q is false only when the antecedent is true and the consequent is false.
If the antecedent is false, then p → q is automatically true.
If the consequent is true, then p → q is automatically true.

12. Example: Determining Whether a Conditional Is True or False
Solution
a) False
Solution
b) True

13. Find the Truth Value of Each Statement
Assume that p and r are false, and q is true.
~p → ( q ˄ r)
(p → ~ q) → (~p ˄ ~ r)
False
True

14. Determine whether the following are True or False…
Antecedent is true, consequent is false, the given is false
Antecedent is False, so the given is true
Consequent is true, so the statement is true

15. Construct a Truth Table
Construct the truth table for (~p → ~q) → (~p ˄ q)
Solution
p q
~ p
~ q
~p → ~q
~p ˄ q
(~p → ~q) → (~p ˄ q)
T T
T F
F T
F F F F T F

16. Construct a Truth Table
Notice
All Truth
Construct a Truth Table
Construct the truth table for (p → q) → (~p ˅ q)
Solution
p q
~ p
p → q
~p ˅ q
(p → q) → (~p ˅ q)
T T
T F
F T
F F T T T T

17. Tautology
A statement that is always true, no matter what the truth values of the components, is called a tautology. They may be checked by forming truth tables.
Other Examples
p ˅ ~p
p → p
(~p ˅ ~ q) → (q ˄ p)

18. Negation of a Conditional
Suppose I say,
“If you get an A, I will bring you cookies”
When do you know for sure you have been lied to?
~ (p → q)
You are misled when
you earned an A and I don’t bring cookies.
p ˄ ~ q
The negation of p → q is p ˄ ~q

19. Construct a Truth Table
Construct the truth table for ~ (p → q) ≡ (p ˄ ~ q)
Solution
p q
p → q
~(p → q)
~ q
p ˄ ~ q
T T
T F
F T
F F F F T T F F F F
The negation of p → q is p ˄ ~q

20. Example: Determining Negations
Determine the negation of each statement.
If you ask him, he will come.
b) All dogs love bones.
Solution
You ask him and he will not come.
b) It is a dog and it does not love bones.

21. Writing a Conditional as an “or” Statement
Since, ~ (p → q) ≡ p ˄ ~ q
Negate ~ [~(p → q)] ≡ ~ (p ˄ ~ q)
Simplify p → q ≡ ~ p ˅ q
Therefore,

22. True or False True False
If the antecedent of a conditional statement is false, the conditional statement is true.
If q is true, then (p˄q)→q is true.
The negation of “If pigs fly, I’ll believe it” is “If pigs don’t fly. I won’t believe it.”
Given that ~p is true and q is false, the conditional p→q is true.
True
False

23. Determine if the following are Logically Equivalent
Any Tautologies?
Is p → q ≡ ~q → ~p ?
Is ~p ˄ ~q ≡ ~q → ~p ? p q ~p ~q
p → q
~q → ~p p q ~p ~q
~p ˄ ~q
~q → ~p
Remember: Statements with matching truth values are logically equivalent

24. Section 3.3
Book Page 118
Circuits

25. Circuits Logic can be used to design electrical circuits.p p q
Series circuit q
p ˄ q
Parallel circuit
p ˅ q
Current will flow through the circuit
when it is closed and not open.

26. Equivalent Statements Used to Simplify Circuits
Remember:
Statements are equivalent when they have the exact same truth table in the final column!

27. Equivalent Statements Used to Simplify Circuits
If T represents any true statement and
F represents any false statement, then

28. Example: Drawing a Circuit for a Conditional Statement
Draw a circuit for
Solution
~ p q
~ r
© 2008 Pearson Addison-Wesley. All rights reserved

29. Assignment Book Page 120: Multiples of 4
© 2008 Pearson Addison-Wesley. All rights reserved