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Math 300
Exam 1 Review (Ch. 2)
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Decide whether ⊆ , ⊂ , both, or neither can be placed in the blank to make a true statement.
1) {0} ∅
A) Both ⊂ and ⊆
B) ⊆
C) Neither
D) ⊂
Explanation:
1)
A)
B)
C) Remember that A⊆B means that every element of A is also in B. So {0} ⊆B is not
true, since 0 is in {0}, but not in ∅. Remember also that A⊂B means that every
element of A is also in B, and that B has at least one element that is not in A. But ∅
has no elements in it, so this clearly does not apply.
D)
Objective: (2.2) Identify Proper Subsets
Describe the conditions under which the statement is true.
2) A ∪ B = B
A) Always true
B) B ⊆ A
Explanation:
2)
C) A ⊆ B
D) A = ∅
A)
B)
C) Remember that A∪B means the set containing all elements from both sets A and B.
In other words, it is the result of combining sets A and B. If the combination of sets
A and B is the same as set B, then all elements of A must already be in B. Thus A
⊆B.
D)
Objective: (2.3) Describe Conditions Under Which Statement Is True
3) A ∩ Aʹ = A
A) Always true
Explanation:
3)
B) A ≠ ∅
C) A = U
D) A = ∅
A)
B)
C)
D) Remember that Aʹ is the complement of A, meaning the set containing everything
in the universal set outside of A. Thus A and Aʹ have no elements in common, so
their intersection is empty. So we know that A ∩Aʹ = ∅. Therefore, the only way
that A∩Aʹ can be equal to A is if A=∅ .
Objective: (2.3) Describe Conditions Under Which Statement Is True
Determine whether the statement is true or false.
Let A = {1, 3, 5, 7}
B = {5, 6, 7, 8}
C = {5, 8}
D = {2, 5, 8}
U = {1, 2, 3, 4, 5, 6, 7, 8}
4) D ⊆ B
A) True
Explanation:
4)
B) False
A) D contains the element 2, which is not in B. Thus D is not a subset of B.
B)
Objective: (2.2) Determine Truth of Statement: Subsets
1
5)
∅ ⊆ A
5)
A) True
B) False
Explanation:
A) Remember that A⊆B means that ʺevery element of A is also an element of B.ʺ This
is equivalent to saying that ʺA contains no elements that are not in B.ʺ Therefore,
the empty set must be a subset of every set. (Note that since ∅ is empty, it
contains no elements that are not in B no matter what B is. In fact, ∅ contains no
elements whatsoever.
B)
Objective: (2.2) Determine Truth of Statement: Subsets
Let A and B be sets with cardinal numbers, n(A) = a and n(B) = b, respectively. Decide whether the statement is true or
false.
6)
6) n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
A) True
B) False
Explanation:
A) Remember that A∪B is the set containing all elements that belong to A or B (or
both). So n(A) + n(B) has counted any elements of A ∩B twice, and thatʹs why we
need to subtract n(A ∩B) to make up for counting them twice.
B)
Objective: (2.3) Determine Truth of Statement: Cardinal Numbers
7) n(A ∩ B) = n(A) - n(B)
A) True
Explanation:
7)
B) False
A)
B) Hereʹs a counterexample: Let A = {1, 2, 3, 4} and B = {1}. Then
n(A∩B) = 1, but n(A) - n(B) = 4 -1 = 3. This proves that the statment is not true in
all cases.
Objective: (2.3) Determine Truth of Statement: Cardinal Numbers
Let U = {all soda pops}, A = {all diet soda pops}, B = {all cola soda pops}, C = {all soda pops in cans}, and
D = {all caffeine-free soda pops}. Describe the set in words.
8) Aʹ ∩ C
A) All diet soda pops and all soda pops in cans
B) All non-diet soda pops in cans
C) All diet soda pops in cans
D) All non-diet soda pops and all soda pops in cans
Explanation:
A)
B) Aʹ∩C means the set of all elements that are in C but not in A. An element being in
C means that it is a soda pop in a can. Not being in A means that it is not a diet
soda pop. So the intersection of these sets is the set of all non-diet soda pops in
cans.
C)
D)
Objective: (2.3) Describe Set in Words
2
8)
List the elements in the set .
Let U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}. 9) B ∩ C
A) {w, y, z}
C) {y}
Explanation:
9)
B) {q, s, v, w, x, y, z}
D) {y, z}
A)
B)
C)
D) B∩C means the set of all elements common to B and C. We can see that y and z are
the only elements that B and C have in common.
Objective: (2.3) Use Set Operations I
10) A ∩ Bʹ
A) {u, w}
C) {r, s, t, u, v, w, x, z}
Explanation:
10)
B) {t, v, x}
D) {q, s, t, u, v, w, x, y}
A) A∩Bʹ means the set of all elements that are in A but not in B. We can see that u and
w are the only elements that fiet this description.
B)
C)
D)
Objective: (2.3) Use Set Operations I
11) B ∩ (A - C)
A) {q, r, s, t, u, v, w, x, y}
C) {q, s, u, y, z}
Explanation:
11)
B) {q, s, u, y}
D) {q, s}
A)
B)
C)
D) B∩(A-C) means the set of all elements that are in B and in A but not in C. We can
see that q and s are the only elements belonging to this set.
Objective: (2.3) Use Set Operations II
Tell whether the statement is true or false.
12) {8} = {x | x is an even counting number between 10 and 16}
A) True
B) False
Explanation:
12)
A)
B) The set of all counting numbers between 10 and 16 is the set {11, 12, 13, 14, 15}.
Objective: (2.1) Determine Truth of Statement: Sets I
Use ⊆ or ⊈ in the blank to make a true statement.
13) {e, n, j} {e, e, n, n, j, j}
A) ⊈
Explanation:
13)
B)
⊆
A)
B) These two sets are equal -- they both contain the elements e, n and j (remember
that listing elements more than once doesnʹt change the set). Therefore, they are
both subsets of each other, so B is the right choice.
Objective: (2.2) Determine Subset Relationships
3
Write the set in set-builder notation.
14) {17, 18, 19, 20}
A) {x | x is an integer less than 21}
C) {17, 18, 19, 20}
Explanation:
14)
B) {x | x is an integer between 16 and 21}
D) {x | x is an integer between 17 and 20}
A) This is {1, 2, 3, ..., 20}
B) Clearly, 17, 18, 19 and 20 are the integers between 16 and 21.
C) This is not written in set builder notation
D) The set described is {18, 19}
Objective: (2.1) Use Set-Builder Notation
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Draw an appropriate Venn diagram and use the given information to fill in the number of elements in each region.
15) n(A) = 48, n(B′) = 60, n(C) = 51, n(A ∩ B) = 20, n(B ∩ C) = 19, n(A ∩ C) = 18,
15)
n(A ∩ B ∩ C) = 12, n(A ∪ B) = 70
Explanation:
Objective: (2.4) Draw Venn Diagram, Label Size of Each Region
Find the indicated cardinal number.
16) Find n(G), given that n(D × G) = 20 and D = {7, 8, 9, 10}.
16)
Explanation: We know that n(D × G) = n(D) × n(G) = 4 × n(G) = 20, so n(G) must be 5.
Objective: (2.3) Find Cardinal Number: Cartesian Products
Find n(A) for the set.
1
1 2
2 3
3
19
19
17) A = , - , , - , , - , ..., , - 2
2 3
3 4
4
20
20
17)
Explanation: n(A) means the number of elements in A. We see that the denominators occur
in pairs, and that there are 20 - 1 = 19 pairs, which comes out to 38 elements.
Objective: (2.1) Find n(A) for Set
Find the Cartesian product.
18) A = {2, 4, 7, 6}
B = {0, 1}
Find B × A.
18)
Explanation: Remember the definition of Caresian product:: A × B = {(x, y)| x∈A and y∈B}
Objective: (2.3) Find Cartesian Product
4
Find the cardinal number of the indicated set. Use the cardinal number formula.
19) If n(A) = 7, n(B) = 15 and n(A ∩ B) = 5, what is n(A ∪ B)?
19)
Explanation: n(A∪B) = n(A) + n(B) - n(A∩B)
Objective: (2.4) Use Formula to Find Cardinal Number
Find the cardinal number of the set.
20) The numbers in the Venn Diagram below represent cardinalities.
20)
Find n(A ∩ Bʹ ∩ C)
Explanation: The expression n(A∩Bʹ∩C) means the number of elements that belong to A and
C, but not B.
Objective: (2.4) Determine Cardinality from Venn Diagram
Find the number of subsets of the set.
21) {math, English, history, science, art}
21)
Explanation: Remember that the number of subsets of and set A is equal to 2 n(A).
Objective: (2.2) Find Number of Subsets
For the given sets, construct a Venn diagram and place the elements in the proper region.
22) Let U = {c, d, g, h, k, u, q} A = {d, h, g, q}
B = {c, d, h, u}
Explanation:
Objective: (2.3) Place Elements in Venn Diagram
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22)
Shade the regions representing the set.
23) (A ∪ B) ∩ (A ∩ B)ʹ
23)
Explanation: The expression (A ∪B) ∩ (A∩B)ʹ means the set of all elements that are in the union
but not in the intersection.
Objective: (2.3) Shade Venn Diagram
Solve the problem.
24) List all possible subsets of the set {m, n}.
24)
Explanation: Remember that ∅ is a subset of any set.
Objective: (2.2) List Subsets of Set
The lists below show five agricultural crops in Alabama, Arkansas, and Louisiana.
Alabama
soybeans (s)
peanuts (p)
corn (c)
hay (h)
wheat (w)
Arkansas
soybeans (s)
rice (r)
cotton (t)
hay (h)
wheat (w)
Louisiana
soybeans (s) sugarcane (n)
rice (r)
corn (c)
cotton (t)
Let U be the smallest possible universal set that includes all of the crops listed, and let A, K and L be the sets of five
crops in Alabama, Arkansas, and Louisiana, respectively. Find each of the following sets.
25) The set of crops in Aʹ.
25)
Explanation: These are the only crops not grown in Alabama.
Objective: (2.2) Solve Apps: Find Complement/Intersection
26) Let A = {(x,y)|x2 +y2 = 25} and B = {(x,y)|y-x=1}. Find A∩B .
Objective:
26) ____________________
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