Topics in Pre Calc 1
Review for Test on Unit 2 Part 2 Name:
Part I: Multiple Choice
1) The statement ~(𝑝 ∧ ~𝑞) is logically equivalent to:
(1) ~𝑝 ∧ 𝑞 (2) ~𝑝 ∨ 𝑞 (3) ~𝑝 → 𝑞 (4) ~𝑝 ∨ ~𝑞
Which law(s) did you use?
2) If the temperature in my room is above 80 degrees, the air conditioner goes on. The air conditioner is not on. Which is a valid conclusion?
(1) The temperature in my room is going up.
(2) The temperature in my room is going down.
(3) The temperature in my room is above 80 degrees.
(4) The temperature in my room is not above 80 degrees.
Which law(s) did you use?
3) Given the premises 𝑎 → 𝑏 and 𝑎, which of the following is a logical conclusion?
(1) 𝑎 (2) ~𝑎 (3) 𝑏 (4) ~𝑏
Which law(s) did you use?
4) Which conclusion logically follows from the true statements “If we do not save fuel, there will be an energy crisis” and “If there is an energy crisis, schools will close”?
(1) If we save fuel, there will not be an energy crisis.
(2) If we do not save fuel, schools will close.
(3) If the schools close, there is an energy crisis.
(4) If we save fuel, there is an energy crisis.
Which law(s) did you use?
5) What is the negation of 𝑝 ∨ ~𝑞?
(1) ~𝑝 ∨ 𝑞 (2) ~𝑝 ∧ 𝑞
Which law(s) did you use?
(3) 𝑝 ∧ ~𝑞 (4) ~𝑝 ∨ ~𝑞

6) If 𝑓 → ~𝑔 and 𝑔 represent true premises, then which statement is true?
(1) ~𝑓 (2) ~𝑔 (3) 𝑓 (4) No conclusion is possible.
If there is a valid conclusion, which law(s) did you use?
7) Which is logically equivalent to the statement “If I eat, then I live”?
(1) If I do not eat, then I do not live.
(2) If I do not live, then I do not eat.
(3) If I live, then I eat.
(4) If I do not eat, then I live.
Which law(s) did you use?
8) If 𝑎 → 𝑏 and ~𝑐 → ~𝑏 then it follows logically that:
(1) 𝑎 → 𝑐 (2) 𝑏 → 𝑎 (3) 𝑐 → 𝑎
Which law(s) did you use?
(4) 𝑐 → 𝑏
9) Which of the following is the negation of the statement “Larry is old and Mario is not here”?
(1) Larry is old and Mario is here.
(2) Larry is not old or Mario is here.
(3) Larry is not old and Mario is not here.
(4) Larry is old or Mario is here.
Which law(s) did you use?
10) Given the true statements “If Paul catches fish today, then he will give me some” and “Paul will give me some fish”, which statement must be true?
(1) Paul will not give me some fish.
(2) Paul will not catch some fish today.
(3) Paul will catch some fish today.
(4) No conclusion is possible.
If there is a valid conclusion, which law(s) did you use?
11) Given the true statement [(𝑝 ∨ 𝑞) ∧ (~𝑞)], which statement must also be true?
(1) 𝑝 (2) ~𝑝 (3) 𝑞 (4) 𝑝 → 𝑞
Which law(s) did you use?

15) 𝑟 ∨ 𝑠
~𝑠
Law(s) used:
16) 𝑡 → ~𝑠
~𝑟 → 𝑠
Law(s) used:
17) ~(𝑠 ∧ ~𝑟)
~𝑟
Law(s) used:
18) 𝑟 ∧ ~𝑡 𝑠 → 𝑡
Law(s) used:
12) Given the true statements :
If the sun shines, I will play tennis.
If I play tennis, I will not study math.
I study math.
Which statement must also be true?
(1) I played tennis. (2) The sun didn’t shine.
(3) I didn’t study math. (4) The sun did shine.
Which law(s) did you use?
Part II: Matching
Directions: For each set of premises on the left, choose the correct conclusion from the list on the right. Indicate the law or laws that you used to determine your answer. You may use an answer more than once.
13) 𝑟 → 𝑠
~𝑠 a) ~𝑟
Law(s) used:
14) 𝑟 → 𝑠 𝑟
Law(s) used: b) 𝑟 c) d) e) f)
~𝑠 𝑠 𝑡 → 𝑟 𝑡 → ~𝑟

Part III: Short Answer
Directions: Write a valid conclusion for each set of premises. If no conclusion is possible, write
“no conclusion”. If there is a valid conclusion, indicate the law(s) used.
19) If it snows this weekend, we will go skiing.
We will not go skiing.
Conclusion:
Law(s) used:
20) It rains in April or flowers will not grow in May.
It did not rain in April.
Conclusion:
Law(s) used:
21) If I play tennis, then I will not have time for dinner.
If I do not have time for dinner, then I will not go swimming.
Conclusion:
Law(s) used:
22) I have money or I will stay home.
I do not have money.
Conclusion:
Law(s) used:
23) Jim will get an A if he passes this test.
Jim passes this test.
Conclusion:
Law(s) used:

30) ~𝑞 ∨ 𝑟
~𝑟
∴ ~𝑞
33) ~(𝑝 ∧ 𝑞)
𝑞
∴ ~𝑝
Part IV: Name the inference law(s) that tells why the argument is valid, or state that the argument is invalid.
24) ~𝑟 → 𝑠 𝑡 → ~𝑠
∴ ~𝑟 → ~𝑡
25) ~𝑥 → 𝑦
~𝑥
∴ 𝑦
26) ~𝑝 ∧ 𝑞
∴ ~𝑝
27) ~(𝑝 ∨ ~𝑞)
∴ ~𝑝 ∨ 𝑞
28) ~𝑟
∴ ~𝑟 ∨ 𝑝
29) 𝑝 → ~𝑟
𝑟
∴ ~𝑝
31) 𝑞 ∨ 𝑟
∴ 𝑟
32) ~𝑟
𝑞
∴ ~𝑟 ∧ 𝑞
34) ~𝑟 → 𝑘
∴ 𝑘 → ~𝑟
35) ~𝑥 → 𝑦
𝑦
∴ ~𝑥