Geometry · Unit 1
Use detachment, syllogism, and the contrapositive. Say when there is no conclusion.
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detachment: p → q, p, ∴ q
syllogism: p → q, q → r, ∴ p → r
contrapositive: p → q, ∼q, ∴ ∼p
Law of Detachment
p → q and p, ∴ q.
If the if-then is true and the hypothesis happens, the conclusion happens.
Example: If the light is green, then you may go. The light is green. Therefore, you may go.
Law of Syllogism
p → q and q → r, ∴ p → r.
Chain the arrows. The conclusion of the first is the hypothesis of the second.
p → q → r is the same as p → r.
Example: If it snows, then school is delayed. If school is delayed, then the concert is postponed. Therefore, if it snows, then the concert is postponed.
Law of Contrapositive
p → q is equivalent to ∼q → ∼p.
If the if-then is true and the conclusion is false, the hypothesis is false.
Example: If a number is divisible by 4, then it is even. 15 is not even. Therefore, 15 is not divisible by 4.
No conclusion
p → q together with q does not give p (that is the converse). p → q together with ∼p does not give ∼q (that is the inverse).
Example: If a figure is a square, then it has four right angles. Figure ABCD has four right angles. No conclusion: it might be a rectangle that is not a square.
Contrapositive error
A common error is to treat the inverse as the Law of Contrapositive. From p → q and ∼p you cannot conclude ∼q.
Example: If it is raining, then the picnic is canceled. It is not raining. You cannot conclude that the picnic is not canceled.
Name the law from symbols, or say it is not a syllogism.
The arrows chain: a to b to c.
Yes.
a → c
Answer: yes; a → c
Both arrows start at a. They do not chain.
No.
Answer: no
Both arrows end at b. They do not chain.
No.
Answer: no
Chain c to a to b.
Yes.
c → b
Answer: yes; c → b
Law of Detachment.
q
Answer: q; Law of Detachment
Law of Contrapositive.
∼p
Answer: ∼p; Law of Contrapositive
The Law of Contrapositive needs ∼q to conclude ∼p.
The student used the inverse. That is a contrapositive error.
No. There is no conclusion.
Answer: no; contrapositive error
Draw a valid conclusion from words, and name the rule. If none, say no conclusion.
p → q and p.
Law of Detachment.
ABCDE has 5 sides.
Answer: ABCDE has 5 sides; Law of Detachment
p → q and q → r.
Law of Syllogism.
If you finish the lab, then you catch the early bus.
Answer: If you finish the lab, then you catch the early bus; Law of Syllogism
p → q and ∼q.
Law of Contrapositive.
Triangle PQR is not a right triangle.
Answer: PQR is not right; Law of Contrapositive
p → q and q.
That is the converse. It is not a law of logic.
No conclusion.
Answer: no conclusion
If a figure is a square, then it has 4 right angles.
∼q is given.
Law of Contrapositive.
ABCD is not a square.
Answer: ABCD is not a square; Law of Contrapositive
p → q and ∼p.
That is the inverse. It is not a law of logic.
No conclusion.
Answer: no conclusion
p → q and ∼p.
The classmate used the inverse and called it the Law of Contrapositive.
That is a contrapositive error. No conclusion.
Answer: no; contrapositive error
Word arguments that look like a chain. Some are syllogisms. Some are not.
p → q
q → r
p → r
The arrows chain. Yes. Law of Syllogism.
Answer: yes; Law of Syllogism
The form is p → q and q → r, so p → r.
The chain is valid, even if a real reeded-edge coin might not be a dime.
Yes, as an argument form (Law of Syllogism).
Answer: yes; the chain is a syllogism
p → q and r → q.
Both arrows end at q. They do not chain.
No.
Answer: no
p → q and q.
No conclusion.
Answer: no conclusion
p → q and ∼q.
Law of Contrapositive.
The printer does not have paper.
Answer: The printer does not have paper; Law of Contrapositive
Chain all three arrows.
If you buy a ticket, then you see the opening number.
Answer: If you buy a ticket, then you see the opening number.
p → q and ∼p.
The argument uses the inverse and treats it like the Law of Contrapositive.
No. That is a contrapositive error. No conclusion.
Answer: no; contrapositive error
Rewrite as if-then as needed, then line the arrows up so each conclusion is the next hypothesis.
After Kai showers, he walks home.
When the pool is open, Kai swims laps.
Kai showers if he swims laps.
Rewrite each statement as if-then. Then put the rows in chain order so the conclusion of one line is the hypothesis of the next.
| Original | Rewritten as if-then |
|---|---|
| When the pool is open, Kai swims laps. | If the pool is open, then Kai swims laps. |
| Kai showers if he swims laps. | If Kai swims laps, then he showers. |
| After Kai showers, he walks home. | If Kai showers, then he walks home. |
Conclusion: If the pool is open, then Kai walks home.
Answer: If the pool is open, then Kai walks home.
When the bell rings, Maya leaves class.
Maya walks to the bus if she leaves class.
After Maya gets a seat, she texts her brother.
Maya gets a seat when she walks to the bus.
If Maya texts her brother, then he meets her at the stop.
Rewrite each statement as if-then. Then put the rows in chain order so the conclusion of one line is the hypothesis of the next.
| Original | Rewritten as if-then |
|---|---|
| When the bell rings, Maya leaves class. | If the bell rings, then Maya leaves class. |
| Maya walks to the bus if she leaves class. | If Maya leaves class, then she walks to the bus. |
| Maya gets a seat when she walks to the bus. | If Maya walks to the bus, then she gets a seat. |
| After Maya gets a seat, she texts her brother. | If Maya gets a seat, then she texts her brother. |
| If Maya texts her brother, then he meets her at the stop. | If Maya texts her brother, then he meets her at the stop. |
Conclusion: If the bell rings, then he meets her at the stop.
Answer: If the bell rings, then he meets her at the stop.
Noah sets a timer if he puts in the pizza.
If the timer beeps, then Noah takes out the pizza.
When Noah turns the oven on, it preheats.
After the oven preheats, Noah puts in the pizza.
When Noah sets a timer, the timer beeps.
Rewrite each statement as if-then. Then put the rows in chain order so the conclusion of one line is the hypothesis of the next.
| Original | Rewritten as if-then |
|---|---|
| When Noah turns the oven on, it preheats. | If Noah turns the oven on, then it preheats. |
| After the oven preheats, Noah puts in the pizza. | If the oven preheats, then Noah puts in the pizza. |
| Noah sets a timer if he puts in the pizza. | If Noah puts in the pizza, then he sets a timer. |
| When Noah sets a timer, the timer beeps. | If Noah sets a timer, then the timer beeps. |
| If the timer beeps, then Noah takes out the pizza. | If the timer beeps, then Noah takes out the pizza. |
Conclusion: If Noah turns the oven on, then he takes out the pizza.
Answer: If Noah turns the oven on, then he takes out the pizza.
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