Math exam

1. (5 points) Let A be the set of letters of the alphabet between g and m. Express A in roster form and then in
set-builder notation.
2. (3 points) Determine whether B, the set of letters in the word latter, and C, the set of letters in the word alert,
are equal, equivalent, both, or neither. Start by writing B and C in roster form.
3. (2 points) For (a) and (b), determine whether each statement is true or false:
(a) {bird, cat, dog,snake} ⊆ {x|x is an animal} (b) {LA} ∈ {BOS, LA, NY, SEA}
4. (2 points) For set A = {9, 10, 11, 12, 13, 14} and set B, the set of all natural numbers between 8 and 14 inclusive,
state all of the following that apply: A = B, A ⊆ B, B ⊆ A, A ⊂ B, B ⊂ A. If none apply, state None.
5. (2 points) Ramona is ordering a frozen yogurt. She can get just the frozen yogurt itself, or she can add any of the
following toppings: boba, chocolate syrup, gummy bears, peanuts, sprinkles, strawberries. How many different
variations of a frozen yogurt are possible?
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Math 120 Exam 1 Summer 2021
6. (8 points) For (a) and (b), let U = {1, 2, 3, …, 12}, A = {2, 3, 5, 7, 11}, B = {1, 2, 3, 10, 11, 12}, and C =
{1, 2, 3, 5, 8}. Determine:
(a) A0 and use it to determine A0 − B
(b) A ∪ B and use it to determine (A ∪ B) ∩ C
7. (3 points) In a survey of restaurant customers, 63 customers ordered french fries or a salad, 38 ordered french
fries, and 15 ordered both french fries and a salad. How many of the customers surveyed ordered a salad?
8. (2 points) Let U be the set of restaurants in Los Angeles, B be the set of barbecue restaurants in Los Angeles,
and T be the set of takeout restaurants in Los Angeles. Describe the set B − T in words.
9. (9 points) Construct a Venn diagram illustrating the following sets:
ˆ U = {a, b, c, d, …, n}
ˆ A = {a, b, e, f, i, j, m, n}
ˆ B = {a, b, d, g, k}
ˆ C = {b, d, f, h, j, l, n}
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Math 120 Exam 1 Summer 2021
10. (7 points) Use a three-ring, eight-region Venn diagram as reference and determine whether (A0 ∪ B) ∩ C
0 =
A0 ∩ (B ∪ C
0
) for all sets A, B, and C. You may assume that A corresponds to regions I, II, IV , and V , B
corresponds to regions II, III, V , and V I, and C corresponds to regions IV , V , V I, and V II.
(A0 ∪ B) ∩ C
0 A0 ∩ (B ∪ C
0
)
Set Regions Set Regions
11. (7 points) An electronics store tracked the purchases of 75 customers on a given day. Construct a Venn diagram
based on the results below.
ˆ 24 purchased a television.
ˆ 19 purchased a cellphone.
ˆ 3 purchased both a television and a cellphone.
Based on the above Venn diagram, answer (a) and (b).
(a) How many customers purchased a cellphone, but
not a television?
(b) How many customers purchased neither a television
nor a cellphone?
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Math 120 Exam 1 Summer 2021
12. (4 points) Show that the set A =
1
5
,
1
9
,
1
13 ,
1
17 , …
has cardinal number ℵ0 by establishing a one-to-one
correspondence between the set of natural numbers and A. Be sure to show the pairing of the general
terms in the sets.
13. (1 point) Write the negation of No apples are oranges.
14. (10 points) For simple statements p: It is summer., q: I am thirsty., and r: I will buy lemonade., write the
following compound statements in words:
(a) (q∧ ∼ r) ↔∼ p
(b) q →∼ (p ∨ r)
15. (9 points) Construct a truth table for p∨ ∼ (q∧ ∼ r).
Indicate the answer column.
p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
16. (10 points) For simple statements h: Derek is a history major. and e: Derek is an English major., write the
following compound statement in symbolic form : It is not true that Derek is neither history major nor an
English major. Then determine the truth value of the compound statement if h is true and e is false.
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Math 120 Exam 1 Summer 2021
17. (16 points) Write the following statement in symbolic
form: If I will go to college then I will get a job, if and
only if I will not go to college or I will not get a job. Label any simple statements used with variables. Then
construct a truth table for it and indicate the answer
column.
T T
T F
F T
F F
18. (7 points) Write the following compound statement in symbolic form: If 5 + 1 = 7, then 5 = 7 − 1. Label
any simple statements used with variables and determine their truth values. Then use these truth values to
determine the truth value of the compound statement.
19. (4 points) Use De Morgan’s laws to write an equivalent statement for My cat has a long tail but does not
have pointy ears.
20. (2 points) Write the inverse of If Laura is tall, then Laura is a dentist.
21. (17 points) Use truth tables to determine whether the statements ∼ [p∧ ∼ (q → r)] and (p →∼ q) ∨ r are
equivalent. Indicate the answer columns.
p q r
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
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Math 120 Exam 1 Summer 2021
22. (12 points) For (a) and (b), translate the argument into symbolic form and determine if the argument is valid
or invalid by comparing the argument to a standard form (state the standard form that is used for comparison).
Clearly define and label any simple statements used with variables.
(a)
I will buy a bicycle or I will rent a car.
I will not rent a car.
∴ I will buy a bicycle.
(b)
If Carl is going to the beach, then it is Sunday.
It is Sunday.
∴ Carl is going to the beach.
23. (4 points) Use deductive reasoning to prove that if you pick any number n, subtract 18 from the number,
multiply the difference by 3, and add 54 to the product, the result is three times the original number.
24. (12 points) For (a) and (b), use an Euler diagram to determine whether the syllogism is valid or invalid.
Clearly define and label any sets or elements used with variables.
(a)
Some chairs are made of wood.
A table is made of wood.
∴ A table is a chair.
(b)
All salmon are fish.
All fish have gills.
∴ All salmon have gills