Math 102 Practice Test 1

Write the following {0, 1, 2, ..., 10} in set-builder notation.

Answer explanation

The set {0, 1, 2, ..., 10} can be expressed in set-builder notation as {x | x is an integer and 0 ≤ x ≤ 10}. This notation defines the set by specifying the properties that its members must satisfy.

Write out the set {x: x is an integer less than 4} in roster notation.

Answer explanation

The set {x: x is an integer less than 4} includes all integers less than 4. In roster notation, this is written as {3, 2, 1, 0, -1, -2, ...}. Thus, the answer is {x: x ∈ Z, x < 4} = {..., -2, -1, 0, 1, 2, 3}.

How many subsets of {a, b, c, d, e, f, h, i} are there? Show how you determined it.

Answer explanation

To find the number of subsets of a set with n elements, use the formula 2^n. Here, n = 8 (elements: a, b, c, d, e, f, h, i). Thus, the number of subsets is 2^8 = 256.

Write each of the following in set builder notation. a. {0, 1, 2, 3, 4, 5, 6, 7} b. {January, February, March, April, May}

Answer explanation

a. {x | x is an integer, 0 ≤ x ≤ 7} b. {m | m is a month, m ∈ {January, February, March, April, May}}. This notation defines the elements based on their properties or membership in a specified set.

Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}, B = {1, 3, 4, 5, 7} C = {7, 8} find: a. A ∩ B b. B′ c. A′ ∪ B′ d. A – B e. (A ∩ C) ∩ (A ∪ B)

Answer explanation

a. A ∩ B = {4}; b. B' = {2, 6, 8, 9}; c. A' ∪ B' = {1, 2, 3, 5, 6, 9}; d. A - B = {2, 6, 8}; e. (A ∩ C) ∩ (A ∪ B) = {8}.

List all the subsets of {t, i, m}.

Answer explanation

The subsets of {t, i, m} include the empty set, individual elements, and combinations: {}, {t}, {i}, {m}, {t, i}, {t, m}, {i, m}, and {t, i, m}. Thus, there are 8 subsets in total.

Cinthia has 237 minutes remaining on her cell phone this month and wants to use them about equal to call her 11 good friends. Show how to determine the number of minutes she can talk to each of her friends.

Answer explanation

To find out how many minutes Cinthia can talk to each friend, divide the total minutes (237) by the number of friends (11). This gives approximately 21.5 minutes per friend.

What does the shaded region in each figure below represent in set notation?

Answer explanation

The shaded region in each figure represents the elements that belong to a specific set or the intersection of sets, typically denoted in set notation as A, B, or A ∩ B, depending on the context of the figures.

Illustrate each of the following by shading the Venn diagrams below. a. S′ ∩ K b. (T ∪ M)’

Answer explanation

For S′ ∩ K, shade the area outside S that overlaps with K. For (T ∪ M)’, shade the area outside the union of T and M, covering everything not in either set.

Use the diagram below to find the cardinality for each problem. a. n(N) = b. n[(R ∩ E) ∪ N] = c. n(E′) = d. n(E – R) = e. n(U) = f. n(R ∩ E ∩ N) =

Answer explanation

To find the cardinalities, count the elements in each set as per the diagram. For example, n(N) counts elements in N, n[(R ∩ E) ∪ N] combines R and E intersections with N, and so on. Ensure to follow set operations correctly.