Warm-Up

IF YOU PAY ATTENTION, THEN YOU WILL LEARN…

IF YOU DON’T PAY ATTENTION, THEN YOU ARE DOOMED…

2.2 Conditional Statements

Introduction to Conditionals

A conditional statement is often referred to as an “implication” or more commonly an “if, then statement”

An example of a conditional statement. If you drink your milk, then you will build strong

bones.

There are two main parts to a conditional Hypothesis (after the if, but before the then)

Conclusion (after the then)

Identify the Hypothesis and Conclusion of each Condition

If you have an eye patch, then you must be a pirate.

If you are a pirate, then you have a wooden leg.

If you have an eye patch, then you have a wooden leg.

If you are a parrot, then you know a pirate.

A person is a pirate, if he has a parrot.

Rewrite each statement as a proper Conditional

Being a good student implies you do your homework. If you’re a good student, then you do your homework.

Being a leprechaun implies you have a pot of gold. If you’re a leprechaun, then you have a pot of gold.

Symbol Form of a Conditional

Conditional will often be written symbolically

Variables will be used to take the place of the hypothesis and conclusion

Ex. of a conditional in symbolic form p → q

p represents the hypothesis, q the conclusionp → q is read as “p implies q” or “if p, then q”

Converse

When we take the converse of a conditional statement we swap the locations of our hypothesis and conclusion (q p)

Ex. If you are a Mogwai, then you shouldn’t be fed after

midnight.

The converse would be… If you shouldn’t be fed after midnight, then you are a

Mogwai.

Write the converse of the following conditionals.

If you are a cheetah, then you have tear lines. If you have tear lines, then you are a cheetah.

If you live on Mars, then you are a Martian. If you are a Martian, then you live on Mars.

If you aren’t nocturnal, then you’re not a possum. If you’re not a possum, then you’re not nocturnal.

Inverse

When using an inverse we negate our hypothesis and conclusion

Ex. If you like balloons, then you’ll love balloon animals.

The inverse would be… If you don’t like balloons, then you won’t love balloon

animals. If you don’t like balloons, then you’ll hate balloon

animals.

We try to rewrite these in proper English sentences.

Write the inverse of the following conditionals.

If you walk on two feet, then you are bipedal. If you don’t walk on two feet, then you are not

bipedal.

If you don’t eat meat, then you aren’t a carnivore. If you eat meat, then are a carnivore.

If you are a penguin, then you can’t fly. If you aren’t a penguin, then you can fly.

Contrapositive

Contrapositive is a combination of inverse and converse

You must swap the hypothesis and conclusion, and you must negate the hypothesis and conclusion.

Ex. If you are extant, then you exist.

The contrapositive is… If you don’t exist, then you aren’t extant. If you don’t exist, then you are extinct.

Write the contrapositive of the following conditionals.

If you are a deer, then you like to frolic. If you don’t like to frolic, then you aren’t a deer.

If you don’t eat mice, then you’re not an owl. If you’re an owl, then you eat mice.

If you are a shrimp, then you’re not jumbo. If you’re jumbo, then you’re not a shrimp.

Symbol form of Inverse, Converse, and Contrapositive

~ is the logic symbol for negation read as “not”

Conditional: p → q

Inverse: ~p → ~q

Converse: q → p

Contrapositive: ~q → ~p

Use the symbols to write the sentences asked for.

p = we are having a partyq = we are getting a clownConditional:

If we are having a party, then we are getting a clown.

Inverse: If we aren’t having a party, then we aren’t getting a clown.

Converse: If we are getting a clown, then we are having a party.

Contrapositive: If we aren’t getting a clown, then we aren’t having a party.

Example

Determine the validity of each statement

Conditional: If today is Wednesday, then tomorrow is

Thursday. Converse:

If tomorrow is Thursday, then today is Wednesday.

 Inverse: If today is not Wednesday, then tomorrow is

not Thursday. Contrapositive:

If tomorrow is not Thursday, then today is not Wednesday

Validity

A conditional statement is equivalent (true or false) to its contrapositive

A converse statement is equivalent (true or false) to its inverse