Symbol

Symbol Name Meaning

p, q, or r

¬

^

v

->

<->

ImplicationsFor two simple propositions p and q, p q

means if p is true, then q is also true.p: it is rainingq: I am carrying my polka dot umbrellap q states: if it is raining then I am carrying

my polka dot umbrella.

Implication ‘if ….. then …..’

Implications are written as and can be read as

p q

• If p then q

• p only if q

• p implies q

• p is a sufficient condition for q

• q if p

• q whenever p

Implications: Truth Table

p q p -> qT T TT F FF T TF F T

Consider the following propositions

p: It is raining

q: I carry an umbrella

If it is raining then I carry an umbrella

:p q

T

T

T

p: It is raining

q: I carry an umbrella

If it is raining then I carry an umbrella

:p q

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

:p q

p: It is raining

q: I carry an umbrella

If it is raining then I carry an umbrella

:p q

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!

TFF

TTF

FFT

TTT

qp p q

F

F

T

The implication is true, as if it is not raining, I am not carrying the umbrella

ImplicationsYes, this can lead to some “nonsense”-

sounding clauses:If (4 < 3) then (75 > 100) is TRUE

Even some theological quandriesIf (1 < 0) then god does not exist is also TRUENote, if you make that “if (1 > 0) …” we can’t

tell!

Determine whether the statement pq is logically true or falseIf 5 * 4 = 20, then the Earth moves around

the sunp 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 pq 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 -> qT T TT F FF T TF F T

ConverseThe 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!!!

INVERSEIf a quadrilateral is a rectangle, then it is

a parallelogram¬p -> ¬qIf a quadrilateral is not a rectangle then it is

not a parallelogram. Negate both propositions

ContrapositiveIf a quadrilateral is a rectangle, then it is

a parallelogram ¬q -> ¬pIf a quadrilateral is not a parallelogram then

it is not a rectangle. Negate both propositions AND change the

orderConverse + 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 classq: Elizabeth is F-4

p q states: If Elizabeth is in her math classroom, then she is in

F-4q 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 equivalentp<-> q

Equivalent PropositionsThe truth value of equivalence is true only when all of

the propositions have the same truth value.

p q p <-> qT T TT F FF T FF F T

T

T

T

T

F

FI 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.

F

T

FI did not buy Norma the Mar bar so I lied and therefore the equivalence statement is false.

TFF

FTF

FFT TTT

qp p q

F

F

T

The equivalence is true as I did not buy Norma a Mars bar and she did not win Crazy 8’s

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

:p q

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

:p q

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

:p q

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

:p q

Creating longer propositionsWhen 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 propositionsConstruct a truth table for

p q

FF

TF

FT

TT

qp

T

F

T

F

q

F

F

T

F

p q

Create a truth table for p q r

FFF

TFF

FTF

FFT

TTF

TFT

FTT

TTT

rqp q r p q r T

F

F

T

F

F

F

F

FFF

FTF

FFT TTT

qp p q

T

F

F

T

F

T

T

T

TFF

TTF

FFT TTT

qp p q

Create a truth table for

T

T

F

F

T

T

T

T

FFF

FTF

FFT TTT

qp p q

T

F

T

T

T

F

T

F

TFF

TTF

FFT

TTT

qp p q

p q r

T

T

F

T

F

T

T

T

T

F

T

T

T

F

T

F

F

F

F

T

F

T

T

T

TF

FT

p p

FFF

FTF

FFT

TTT

qp p q

FFF

TTF

TFT

TTT

qp p q

TFF

FTF

FFT

TTT

qp p q

TFF

TTF

FFT

TTT

qp p q

Negation Conjunction Disjunction

EquivalenceImplication

Translating English Sentences

p p qp q p q p q

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

p q

x is not an even number or a prime number

p: x is a even number

q: x is a prime number

p q

It is not raining

p: It is raining p

Jesse and Savanna both did the IB Test

p: Renzo did the IB Test

q: Rafael did the IB Test

p q

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

p q

p q p q

p q p

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. Therefore I went to bed late.

p: I go to bed late

q: I feel tired

p q

p q q p

p q q

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. Therefore I earn money.

p: I earn money

q: I go to work

p q

p q q p

p q q

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 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 Diplomap q

p q q r

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

p q q r p

p q q r p r

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.

Logical EquivalenceThere 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 equivalentEX: ¬p V ¬q and ¬(p^q)

p q p ^ q ¬p ¬q

T T TT F FF T FF F F

TF

FT

p p

FFF

FTF

FFT

TTT

qp p q

FFF

TTF

TFT

TTT

qp p q

TFF

FTF

FFT

TTT

qp p q

TFF

TTF

FFT

TTT

qp p q

Negation Conjunction Disjunction

EquivalenceImplication

TautologyA 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

ContradictionA 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.

p p q

• 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

In order to show that is

a tautology we must create a truth table

p p q

p q

T T

T F

F T

F F

p p q p p q F

F

T

T

TFF

TTF

FFT

TTT

qp p q

T

F

T

T

T

T

T

T

The statement is true for all truth values given to p and q

p p q Therefore is logically valid.

TF

FT

p p

FFF

FTF

FFT

TTT

qp p q

FFF

TTF

TFT

TTT

qp p q

TFF

FTF

FFT

TTT

qp p q

TFF

TTF

FFT

TTT

qp p q

Negation Conjunction Disjunction

EquivalenceImplication