1. Logic Day #2 Math Studies IB NPHS Miss Rose

2. Symbol
Symbol Name
Meaning
p, q, or r ^ v
->
<->

3. Implications
For two simple propositions p and q, p  q means if p is true, then q is also true.
p: it is raining
q: I am carrying my polka dot umbrella
p  q states: if it is raining then I am carrying my polka dot umbrella.

4. Implications are written as and can be read as
Implication ‘if ….. then …..’
Implications are written as and can be read as
If p then q
p implies q
p only if q
p is a sufficient condition for q
q if p
q whenever p

5. Implications: Truth Tableq
p -> q T F

6. Consider the following propositions p: It is raining
q: I carry an umbrella
If it is raining then I carry an umbrella T F q p T T T
p: It is raining
q: I carry an umbrella
If it is raining then I carry an umbrella T F F
The implication is false as it is raining and I am not carrying an umbrella
p: It is raining
q: I carry an umbrella
If it is raining then I carry an umbrella F T T
The implication is true, as if it is not raining, I may still be carrying my umbrella. Maybe I think it will rain later, or maybe I am going to use it as a defensive weapon!
p: It is raining
q: I carry an umbrella
If it is raining then I carry an umbrella F F T
The implication is true, as if it is not raining, I am not carrying the umbrella

7. Implications Yes, this can lead to some “nonsense”-sounding clauses:
If (4 < 3) then (75 > 100) is TRUE
Even some theological quandries
If (1 < 0) then god does not exist is also TRUE
Note, if you make that “if (1 > 0) …” we can’t tell!

8. Determine whether the statement pq is logically true or false
If 5 * 4 = 20, then the Earth moves around the sun
p is true (5 *4 does = 20)
q is true (The Earth does revolve around the sun)
SO p  q is logically true!
“If NPHS is the Panthers, then Allison is an alien”
p is true (NPHS is the Panthers)
q is false (Allison..?)
SO p  q is logically FALSE!

9. Determine whether the statement pq is logically true or false
“If Miss Rose has red hair, then Axel is the president”
p is false (Miss Rose does not have red hair)
q is false but that doesn’t matter!!!!
SO p  q is logically true! p q
p -> q T F

10. Converse The converse is the reverse of a proposition.
The converse of p  q is q  p
p  q states: if it is raining then I am carrying my polka dot umbrella.
q  p states: if I am carrying my polka dot umbrella then it is raining.
Even if the implication is true, the converse is not necessarily true!!!

11. INVERSE
If a quadrilateral is a rectangle, then it is a parallelogram
¬p -> ¬q
If a quadrilateral is not a rectangle then it is not a parallelogram.
Negate both propositions

12. Contrapositive ¬q -> ¬p
If a quadrilateral is a rectangle, then it is a parallelogram
¬q -> ¬p
If a quadrilateral is not a parallelogram then it is not a rectangle.
Negate both propositions AND change the order
Converse + Inverse = Contrapositive

13. Equivalent Propositions:
If two combined propositions are true and converse, they are said to be equivalent propositions.
p: Elizabeth is in her math class
q: Elizabeth is F-4
p  q states:
If Elizabeth is in her math classroom, then she is in F-4
q  p states:
If Elizabeth is in F-4, then she is in her math classroom
The two combined statements are both true and converse so they are said to be equivalent
p<-> q

14. Equivalent Propositions
The truth value of equivalence is true only when all of the propositions have the same truth value. p q
p <-> q T F

15. Consider the following propositions p: I will buy Norma a Mars bar
q: She wins the game of Crazy 8’s
I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s T F q p T T T
p: I will buy Norma a Mars bar
q: She wins the game of Crazy 8’s
I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s T F F
I brought her the Mars bar even though she didn’t win the game of Crazy 8’s I lied…so the equivalence statement is false.
p: I will buy Norma a Mars bar
q: She wins the game of Crazy 8’s
I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s F T F
I did not buy Norma the Mar bar so I lied and therefore the equivalence statement is false. F
p: I will buy Norma a Mars bar
q: She wins the game of Crzy 8’s
I will buy Norma a Mars bar if and only if she wins a game of Crazy 8’s F T
The equivalence is true as I did not buy Norma a Mars bar and she did not win Crazy 8’s

16. Creating longer propositions
When creating truth tables for long propositions, always move from simple  complex
Start with the truth values of each simple proposition
then state any negations
then begin working on compound propositions.

17. Creating longer propositions
Construct a truth table for F T q p

18. Create a truth table forq p F T q p
Create a truth table for F T r q p T T F F F F T T F F F T F T F T

19. Create a truth table for p q r

20. If you know each of these, you can do any truth table!
Negation
Conjunction
Disjunction T F p F T q p F T q p
If you know each of these,
you can do any truth table!
Implication
Equivalence T F q p T F q p

21. Translating English Sentences
not p
it is not the case that p
p and q
p or q
if p then q
p implies q
if p, q
p only if q
p is a sufficient condition for q
q if p
q whenever p
q is a necessary condition for p
p if and only if q

22. If there is a thunderstorm then Allison cannot use the computer
p: There is a thunderstorm
q: Allison uses the computer
x is not an even number or a prime number
p: x is a even number
q: x is a prime number

23. It is not raining
p: It is raining
Jesse and Savanna both did the IB Test
p: Renzo did the IB Test
q: Rafael did the IB Test

24. If it is raining then I will stay at home. It is raining
If it is raining then I will stay at home. It is raining. Therefore I stayed at home.
p: It is raining
q: I stay at home
If it is raining then I will stay at home
If it is raining then I will stay at home. It is raining
If it is raining then I will stay at home. It is raining. Therefore I stayed at home.

25. If I go to bed late then I feel tired. I feel tired
If I go to bed late then I feel tired. I feel tired. Therefore I went to bed late.
p: I go to bed late
q: I feel tired
If I go to bed late then I feel tired.
If I go to bed late then I feel tired. I feel tired.
If I go to bed late then I feel tired. I feel tired. Therefore I went to bed late.

26. I earn money if and only if I go to work. I go to work
I earn money if and only if I go to work. I go to work. Therefore I earn money.
p: I earn money
q: I go to work
I earn money if and only if I go to work.
I earn money if and only if I go to work. I go to work.
I earn money if and only if I go to work. I go to work. Therefore I earn money.

27. If I study then I will pass my IB Mathematics
If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study. Therefore I will get my IB Diploma.
p : I Study
q : I will pass my IB Mathematics
r : I will get my IB Diploma
If I study then I will pass my IB Mathematics
If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma
If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study.
If I study then I will pass my IB Mathematics. If I pass my IB Mathematics I will get my IB Diploma. I study. Therefore I will get my IB Diploma.

28. THE END!!!

29. Logical Equivalence p q p ^ q ¬p ¬q T F
There are many different ways to form compound statements from p and q.
Some of the different compound propositions have the same truth values.
In that case, the compound propositions are logically equivalent
EX: ¬p V ¬q and ¬(p^q) p q
p ^ q ¬p ¬q T F

30. Negation
Conjunction
Disjunction T F p F T q p F T q p
Implication
Equivalence T F q p T F q p

31. Tautology
A tautology is a compound proposition that is always true regardless of the individual truth values of the individual propositions.
A compound proposition is valid if it is a tautology

32. Contradiction
A contradiction is a compound proposition that is always false regardless of the individual truth values of the individual propositions

33. Show that the statement is logically valid.
To show that a combined proposition is logically valid, you must demonstrate that it is a tautology.
A tautology is a statement that always tells the truth

34. p q T F In order to show that is
a tautology we must create a truth table p q T F F T T F F T T T T T T T
The statement is true for all truth values given to p and q
Therefore is logically valid.

35. Negation
Conjunction
Disjunction T F p F T q p F T q p
Implication
Equivalence T F q p T F q p