Section 1.3 Implications

Vocabulary Words conditional operation ( ⇒ or →) conditional proposition conditional statement (implication statement) hypothesis (antecedent) conclusion (consequent) converse contrapositive inverse biconditional operation ( ⇔ or ↔)

One more time… Last time I showed the following order of precedence: ¬highest   lowest And today we will add in these even lower , 

But… But, we discussed that some sources say: ~highest ,  ,  lowest

Who is “right” I searched several discrete textbooks on my bookshelf and found nearly a 50/50 split

Who is “right” Several programming languages agree with your textbook (Python, Java, C++) But other’s disagree (Smalltalk, Ruby, and, to some extent, Ada)

Who is “right” Even wikipedia can’t decide… Wikipedia - First Order LogicWikipedia - Logical Connectives

The bottom line(s) I will follow your book’s rules to keep things consistent. BUT, use parentheses to clear up confusion. AND, pay attention when you pick up a new language

Conditional Statements (aka Implication) “If I go to Fareway I will buy Diet Mountain Dew” This sort of statement is known as a conditional statement (or an implication statement).

Conditional/Implication In logic this is written in the form: p  q And we read this as: –If p then q –p implies q (“Going to Fareway implies that I will buy Diet Mountain Dew”)

Conditional/Implication In logic this is written in the form: p  q We state that p is the hypotheses or the antecedent (or assumption or premise) We state that q is the conclusion or the consequent

Conditional/Implication The original statement is False when p is true and q is false; otherwise it is true. pq p  q TTT TFF FTT FFT

Conditional/Implication A conditional statement that is true by virtue of the hypothesis being false is called vacuously true or true by default. (This situation always messes with students) pq p  q TTT TFF FTT FFT

Logical Equivalence: Conditional/Implication Notice that ( p  q)  ~p  q pq p  q ~p  q TTTT TFFF FTTT FFTT

In-Class Activity #1 If you study in this course you will get an A. Tomorrow is Friday if today is Thanksgiving. n is prime implies n is odd or n is 2. Tim is Ann’s father is sufficient for Jim being her uncle and Sue being her aunt. n is divisible by 6 only if n is divisible by 2 and n is divisible by 3. P being a rectangle is necessary for P being a square.

In-Class Activity #1 If you study in this course you will get an A. –Study  Get an A Tomorrow is Friday if today is Thanksgiving. –Today is Thanksgiving  Tomorrow is Friday n is prime implies n is odd or n is 2. –N is prime  n is odd or n is 2

In-Class Activity #1 Tim is Ann’s father is sufficient for Jim being her uncle and Sue being her aunt. –Tim is Ann’s father  Jim is her uncle and Sue is her aunt n is divisible by 6 only if n is divisible by 2 and n is divisible by 3. –N is divisible by 6  n is divisible by 2 and n is divisible by 3 P being a rectangle is necessary for P being a square. –P is a square  P is a rectangle

Order of Precedence ¬highest   ,  lowest

In-Class Activity #2 pq ¬ p  q  ¬q TT(1) TF(2) FT(3) FF(4)

What is the negation of an implication statement How do we write: ¬( p  q) Remember that: ( p  q)  ¬ p  q Therefore: ¬( p  q)  ¬ p  q)  p  ¬ q

Negation of an implication ¬( p  q)  p  ¬ q What is the negation of “If I go to Fareway I will buy Diet Mountain Dew” I will go to Fareway but I will NOT buy Diet Mountain Dew Notice that this does NOT start with an if.

Contrapositive of an Implication The contrapositive of the conditional statement “if p then q” is written as “if not q then not p” The contrapositive of p  q is written as ¬ q  ¬ p

So what… Is a conditional statement logically equivalent to it’s contrapositive? Build the truth table and you will see that it is!

The Converse of an Implication The converse of the conditional statement “if p then q” is written as “if q then p” The converse of p  q is written as q  p

So what… Is a conditional statement logically equivalent to it’s converse? Build the truth table and you will see that it is not! However, this is one of the most common logic errors made by beginning students.

The Inverse of an Implication The inverse of the conditional statement “if p then q” is written as “if not p then not q” The inverse of p  q is written as ¬ p  ¬ q

So what… Is a conditional statement logically equivalent to it’s inverse? Build the truth table and you will see that it is not! HOWEVER, the converse and the inverse are logically equivalent. –One is the contrapositive of the other

In-Class Activity #3 If my client is guilty, then the knife was in the drawer Write the: –Converse –Contrapositive –Inverse –Negation

Converse q  p If the knife was in the drawer then my client is guilty.

Contrapositive ¬ q  ¬ p If the knife was not in the drawer then my client is not guilty.

Inverse ¬ p  ¬ q If my client is not guilty than the knife is not in the drawer.

Negation p ^ ¬ q My client is guilty but the knife is not in the drawer. Notice that the negation of implication is NOT an implication.