- Topics At a Glance
- Building Mathematical Statements
- Negations
- Conjunctions
- Disjunctions
**Conditional Statements**- Converse, Inverse, and Contrapositive
**Detachment and Syllogism**- The Quality of Equality
- Properties of Equality
- Arithmetic Properties
- Proofs
- Formal Proofs
- Postulates and Theorems
- Algebraic Proofs
- Geometric Proofs
- Congruence, Equality, and Geometry

Those words sound awfully fancy, don't they? It should be illegal to say them without a top hat and a monocle. They give two answers to the question, "So, what good are conditional statements, anyway?" They are used as a way of getting new information from information we already have.

The law of **detachment** allows you to "detach" the hypothesis from the conclusion. More precisely, if we know both *p* and *p* → *q* to be true, then we may conclude that *q* is true. When the traffic law says, "If the red light is blinking, then come to a full stop," and you see a blinking red light up ahead, it's clear what you'll do next (hint: resist the temptation to slam the gas pedal).

What can you infer from the following two statements: "All chickens hatch from eggs" and "Betsy is a chicken."

We can write the two statements in shorthand as follows: "chicken → hatch from egg" and "Betsy is a chicken." Since Betsy satisfies the hypothesis of chickenhood, she must by detachment meet the conclusion of egginess. In other words, "Betsy hatched from an egg."

The law of **syllogism**, on the other hand, allows us to squeeze together conditional statements. If we know both *p* → *q* and *q* → *r* to be true, we can squeeze them together to get *p* → *r*. After all, going through *q* when we can go straight from *p* to *r* would be just plain silly-gism.

"If you make a right turn, you must use a turn signal," and, "There is a right turn on the way to school." What can be inferred from these two statements?

As usual, it's a good idea to write the statements in their shorthand to make the structure of the implications clear. The first becomes "right turn → signal" and the second "school → right turn." (That is, if you go to school, then you must make a right turn.) Syllogism lets us write the chain "school → right turn → signal" and then cut out the middle part, leaving us with "school → signal." Translating out of our shorthand, "If you are driving to school, then you will use a turn signal."

We'll list the two laws one last time:

- Detachment: from
*p*→*q*and*p*you may infer*q*. - Syllogism: from
*p*→*q*and*q*→*r*you may infer*p*→*r*.

Example 1

"Calicos are cats" and "All cats are cute." What can be inferred from these statements? |

Example 2

Given the statements, "There is no school during the weekend," and "Bob does his homework during the weekend," what can be inferred? |

Example 3

"All cats are cute" and "All dogs are cute." What can be inferred from these statements? |

Example 4

"Parties are fun," and "Birthdays should have parties." What can be inferred from the inverse of these statements? |

Exercise 1

Given the statements, "Planners are helpful," and, "Calendars are types of planners," what can be inferred?

Exercise 2

"The best flavor of ice cream is chocolate," and "You should always choose the best ice cream flavor." What do these phrases translate to?

Exercise 3

What can you infer from the following statements? "The Beatles made good music," and, "Good music is loved by all."

Exercise 4

What can be inferred from the statements, "Playing classical piano is challenging," and "Challenging activities are rewarding"?

Exercise 5

What do the statements, "Doctor Who is an amazing TV show," and "Whovians are people who watch Doctor Who," translate to?

Exercise 6