Question

for each sentence: a) identify the hypothesis \p\ and b) the conclusion \q\ example: if it rains, then the game is cancelled. a) p: it rains b) q: the game is cancelled 1. if it is 9:00 a.m., then im late to class. a) p: b) q: 2. when it rains, then i do not have to water the lawn. a) p: b) q: 3. you can get to the stadium if you take the third avenue bus. a) p: b) q: 4. the perimeter of a square is 4x + 8 if one side of the square is x + 2. a) p: b) q: 5. if a polygon has exactly three sides, it is a triangle. a) p: b) q: write each sentence in symbolic form, using the propositions below and standard logical connectives (¬,∧,∨,→,↔). • p: \the test is easy.\ • q: \sam studies.\ • r: \sam passes the test.\ 6. if the test is easy, then sam will pass the test. 7. if sam studies, then sam will pass the test. 8. if the test is not easy, then sam will pass the test.

Explanation:

Step1: Recall conditional statement structure

In an "if - then" statement, the part after "if" is the hypothesis $P$ and the part after "then" is the conclusion $Q$.

Step2: Analyze sentence 1

The sentence is "If it is 9:00 a.m., then I'm late to class". So, $P$: It is 9:00 a.m.; $Q$: I'm late to class.

Step3: Analyze sentence 2

The sentence is "When it rains, then I do not have to water the lawn". So, $P$: It rains; $Q$: I do not have to water the lawn.

Step4: Analyze sentence 3

The sentence is "You can get to the stadium if you take the Third Avenue bus". So, $P$: You take the Third Avenue bus; $Q$: You can get to the stadium.

Step5: Analyze sentence 4

The sentence is "The perimeter of a square is $4x + 8$ if one side of the square is $x + 2$". So, $P$: One side of the square is $x + 2$; $Q$: The perimeter of a square is $4x+8$.

Step6: Analyze sentence 5

The sentence is "If a polygon has exactly three sides, it is a triangle". So, $P$: A polygon has exactly three sides; $Q$: It is a triangle.

Step7: Analyze sentence 6

The sentence "If the test is easy, then Sam will pass the test" in symbolic form is $P
ightarrow R$.

Step8: Analyze sentence 7

The sentence "If Sam studies, then Sam will pass the test" in symbolic form is $Q
ightarrow R$.

Step9: Analyze sentence 8

The sentence "If the test is not easy, then Sam will pass the test" in symbolic form is $
eg P
ightarrow R$.

Answer:

1. a) $P$: It is 9:00 a.m.

b) $Q$: I'm late to class

1. a) $P$: It rains

b) $Q$: I do not have to water the lawn

1. a) $P$: You take the Third Avenue bus

b) $Q$: You can get to the stadium

1. a) $P$: One side of the square is $x + 2$

b) $Q$: The perimeter of a square is $4x + 8$

1. a) $P$: A polygon has exactly three sides

b) $Q$: It is a triangle

1. $P

ightarrow R$

1. $Q

ightarrow R$

1. $

eg P
ightarrow R$