Logic Day #2 Math Studies IB NPHS Miss Rose

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

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.

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

Implications: Truth Table q p -> q T F

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

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!

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!

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

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!!!

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

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

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

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

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

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.

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

Create a truth table for q 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

Create a truth table for p q r

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

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

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

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

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.

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.

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.

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.

THE END!!!

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

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

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

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

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

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.

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