Set Theory

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For the Classic ACT exam:
The ACT Mathematics test is a timed exam...60 questions in 60 minutes
This implies that you have to solve each question in one minute.
Each of the first 20 questions (less challenging) will typically take less than a minute a solve.
Each of the next 20 questions (medium challenging) may take about a minute to solve.
Each of the last 20 questions (more challenging) may take more than a minute to solve.
The goal is to maximize your time.
You use the time saved on the questions you solve in less than a minute to solve questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for a wrong answer.
Also: please note that unless specified otherwise, geometric figures are drawn to scale. So, you can figure out the correct answer by eliminating the incorrect options.
Other suggestions are listed in the solutions/explanations as applicable.

These are the solutions to the ACT past questions on the topic: Sets.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided for some questions.
If time permits and there is sufficient interest, video solutions may be provided. Please subscribe to the YouTube channel to be notified of upcoming livestreams. You are welcome to ask questions during the video livestreams.
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(2.) Event A consists of 6 simple events.
Event B consists of 3 simple events, none of which are in Event A.
Event C is the union of A and B, and Event D is the intersection of A and B.
Which of the following statements is true?
(Note: Given that Event X consists of n simple events, |X| = n.)

F. |B| < |A| < |C| < |D|
G. |B| < |A| < |D| < |C|
H. |C| < |B| < |A| < |D|
J. |D| < |B| < |A| < |C|
K. |D| < |C| < |B| < |A|

Compare the events to sets
Let us write examples of the events (sets) based on the question

$ A = \{1, 2, 3, 4, 5, 6\} \\[3ex] |A| = n(A) = 6 \\[5ex] B = \{7, 8, 9\} \\[3ex] |B| = n(B) = 3 \\[5ex] C = A \cup B \\[3ex] C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \\[3ex] |C| = n(C) = 9 \\[5ex] D = A \cap B \\[3ex] D = \phi \\[3ex] |D| = n(D) = 0 \\[5ex] \implies \\[3ex] |D| \lt |B| \lt |A| \lt |C| $

(3.) At a local pet store, 50 shoppers were polled to see if they owned cats or dogs.
Among the polled shoppers, 31 owned at least 1 dog, 20 owned at least 1 cat, 7 owned at least 1 dog and at least 1 cat, and 6 owned neither a dog nor a cat.
How many of the 50 polled shoppers owned at least 1 dog but did NOT own at least 1 cat?

$ F.\;\; 11 \\[3ex] G.\;\; 13 \\[3ex] H.\;\; 23 \\[3ex] J.\;\; 24 \\[3ex] K.\;\; 25 \\[3ex] $

Let the number of shoppers who own at least 1 cat = C
Let the number of shoppers who own at least 1 dog = D
Let us represent the information on a Venn Diagram

Number 3

Those who own at least 1 dog but did NOT own at least 1 cat are those who own only dog = 31 − 7 = 24 shoppers.

(11.) Maya’s digital music library has a total of 249 songs.
Her library has 64 songs that are remixes and 85 hip-hop songs.
Of the hip-hop songs in her library, 25 are remixes.
How many songs in her library are NEITHER remixes NOR hip-hop songs?

$ A.\;\; 75 \\[3ex] B.\;\; 100 \\[3ex] C.\;\; 125 \\[3ex] D.\;\; 139 \\[3ex] E.\;\; 224 \\[3ex] $

Let:
the number of remixes = R
the number of hip-hop somgs = H

Let us represent the information on a Venn Diagram

Number 11

$ 39 + 60 + 25 + n(Neither) = 249 \\[3ex] n(Neither) = 249 - 39 - 60 - 25 \\[3ex] n(Neither) = 125 $

(15.) Veronica delivers 27 copies of the News Report and 22 copies of the City Times to 38 of the 40 houses on Oakland Street.
No house receives more than 1 copy of each newspaper.
How many houses receive both newspapers?

F. 2
G. 5
H. 7
J. 9
K. 11

Let:
News Report = NR
City Times = CT
the number of houses that receive both newspapers = x

Let us represent the information using a Venn Diagram

Number 15

$ 27 - x + x + 22 - x + 2 = 40 \\[3ex] 51 - x = 40 \\[3ex] 51 - 40 = x \\[3ex] x = 11 $

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