Midterm CHP 3-5

The figure below shows a directed graph G: Which edge is not in G+? (3, 3) (4, 1) (2, 3) (1, 4)

(1, 4)

The figure below shows a directed graph G: Which edge is not in G3? (2, 3) (3, 4) (1, 3) (4, 3)

(3, 4)

Select the value of ⌈-5.8⌉ 5 -5 6 -6

-5

f(x)=x2 g(x)=⌊x/2⌋ Select the correct value for f∘g(-3/2)-1 3 1 -1 0

0

f:{0,1}4→{0,1}4. f(x) is obtained by removing the second bit from x and placing the bit at the end of the string. For example, f(1011) = 1110. Select the correct value for f-1(0101). 0011 0110 0101 1010

0110

Graph G is defined by the arrow diagram below. Select the pair of vertices such that there is no walk of length 4 in G from the first vertex to the second vertex. 1, 3 1, 4 2, 1 4, 3

1, 4

Define strings s = 101 and t = 10. Select the string that is equal to st. 10 10110 101 101101

10110

Define strings s = 101 and t = 10. Select the string that is equal to st. 101101 10 10110 101

10110

C={3, 5, 9, 12, 15, 16}D={5, 7, 8, 12, 13, 15} Select the set corresponding to C⊕D. {3, 5, 7, 8, 9, 12, 13, 15, 16} {5, 12, 15} {3, 9, 16} {3, 7, 8, 9, 13, 16}

3, 7, 8, 9, 13, 16}

Select the expression that is equal to log5⁡�3log5⁡27 3log5⁡(�3) log27⁡� log5⁡� log3⁡�

3log5⁡(�3)

A = {1, 2, {3, 4}, {5, 6, 7}} Select the correct value for |A|. 4 7 5 6

4

A = {1, 2, {3, 4}, {5, 6, 7}} Select the correct value for |A|. 5 4 7 6

4

Select the value of ⌊4.2⌋ 4 0 4.2 5

4

Select the value that is equal to ⌊log229⌋ 5 2 4 3

4

The string x is equal to 1101. What is the length of λx? 4 6 3 5

4

The string x is equal to 1101. What is the length of λx? 5 4 6 3

4

A person's birth date consists of the month, day, and year in which that person was born. The domain for a relation R is a set of people. There are at least two people in the group with the same birth date and at least two people with different birth dates. A person x is related to person y under the relation if they have the same birth date or if x's birth date is earlier than y's birth date. Which description correctly characterizes the relation? A partial order A strict order and a total order Neither a partial order nor a strict order. A strict order

??? Not - A strict order and a total order

A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. The matrix A2 is given below. Which vertex can be reached by a walk of length 4 that starts at vertex 1? 2 4 5 3

??? Not 2

f:{0,1}4→{0,1}4. f(x) is obtained from x by replacing the last bit with 1. For example, f(1000) = 1001. Select the correct description of the function f. One-to-one and onto Onto but not one-to-one One-to-one but not onto Neither one-to-one nor onto

??? Not Onto but not one-to-one

Select the collection of sets that forms a partition of Z. Z,Z- Z+,Z- Z,Z-, {0} Z+,Z-, {0}

??? Not Z+,Z-

Which relation on the set {1, 2, 3, 4} is a partial order? { (1, 2), (2, 3), (1, 3), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4) } { (1, 2), (2, 3), (1, 3), (4, 3) } { (1, 2), (2, 3), (1, 3), (3, 4) } { (1, 2), (2, 3), (1, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4, 4) }

??? Not { (1, 2), (2, 3), (1, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4, 4) }

A = {x ∈ Z: x is a prime number} B = {4, 7, 9, 11, 13, 14} C = {x ∈ Z: 3 ≤ x ≤ 10} Select the set corresponding to (A∪B)∩C. {3, 4, 7, 9} {3, 4, 5, 7, 9, 11, 13} {3, 5, 7} {3, 4, 5, 7, 9}

??? Not {3, 5, 7}

A = {1, 2, 3, 4}. Select the statement that is false. ∅∈P(A) ∅⊆P(A) {2,3}∈P(A) {2,3}⊆P(A)

??? Not ∅∈P(A)

A person's birth date consists of the month, day, and year in which that person was born. The domain for a relation R is a set of people. No two people in the group have the same birth date. A person x is related to person y under the relation if x's birth date is earlier than y's birth date. Which description correctly characterizes the relation? A partial order and a total order. A strict order and a total order. A partial order but not a total order. A strict order but not a total order.

A strict order and a total order.

Graph G is defined by the arrow diagram below. Select the properties that accurately describe the following sequence with respect to graph G: ⟨2, 3, 1, 3, 4⟩ A walk but not a trail A trail but not a path A trail and a path Not a walk

A trail but not a path

Select the set that is equivalent to C∪(C∩B). B∩C C∪B ∅ C

C

A={x∈Z:x is even} C={3,5,9,12,15,16} Select the true statement. C - A = {12, 16} The set C - A is infinite. C - A = {3, 5, 9, 12, 15} C - A = {3, 5, 9, 15}

C - A = {3, 5, 9, 15}

Select the law that establishes that the two sets below are equal. �∩(�∪�)¯=�∩(�¯∩�¯) Associative law De Morgan's law Absorption law Distributive law

De Morgan's law

Select the law that establishes that the two sets below are equal. (A∩B)∪(A∩B)=A∩B Absorption law Idempotent law Distributive law Identity law

Idempotent law

Select the law that establishes that the two sets below are equal. (A∩B)∪(A∩B)=A∩B Absorption law Idempotent law Identity law Distributive law

Idempotent law

The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y|≥2. Select the description that accurately describes relation R. Anti-reflexive Reflexive Neither reflexive nor anti-reflexive Transitive

Neither reflexive nor anti-reflexive

f:{0,1}4→{0,1}3. f(x) is obtained from x by removing the first bit. For example, f(1000) = 000. Select the correct description of the function f. Onto but not one-to-one One-to-one but not onto One-to-one and onto Neither one-to-one nor onto

Onto but not one-to-one

The domain of relation R is Z x Z. (a, b) is related to (c, d) if a-b=c-d. Which statement correctly characterizes the relation R? R is not an equivalence relation because R is not reflexive. R is not an equivalence relation because R is not transitive. R is not an equivalence relation because R is not symmetric. R is an equivalence relation.

R is an equivalence relation.

The domain of relation R is the set of all integers. x is related to y if |x-y|≤1. Which statement correctly characterizes the relation R? R is not an equivalence relation because R is not reflexive. R is not an equivalence relation because R is not symmetric. R is an equivalence relation. R is not an equivalence relation because R is not transitive.

R is not an equivalence relation because R is not transitive.

S and T are relations on the real numbers and are defined as follows: S={(x,y)∣x<y}T={(x,y)∣x>y} What is T∘S? T R x R (all pairs of real numbers) ∅ S

R x R (all pairs of real numbers)

S and T are relations on the real numbers and are defined as follows: S={(x,y)∣x<y}T={(x,y)∣x>y} What is T∘S? ∅ S R x R (all pairs of real numbers) T

R x R (all pairs of real numbers)

A is a finite non-empty set. The domain for relation R is the power set of A. (Recall that the power set of A is the set of all subsets of A.) For X⊆A and Y⊆A, X is related to Y if X and Y have the same cardinality (i.e., |X|=|Y|). Select the description that accurately describes relation R. Symmetric and Reflexive Anti-symmetric and Anti-reflexive Symmetric and Anti-reflexive Anti-symmetric and Reflexive

Symmetric and Reflexive

What is the third row of A⋅B? A= | 0 1 0 | | 1 1 0 | | 0 1 0 | B= | 1 1 1 | | 0 1 1 | | 1 0 1 | [0 1 1] [1 1 0] [0 1 0] [1 1 1]

[0 1 1]

A = {a, b, c, d}X = {1, 2, 3, 4} Select the definition for f that is a well-defined function from A to X. f = {(a, 2), (b, 3), (c, 3), (d, 1)} f = {(a, 2), (b, 3), (c, 3), (1, d)} f = {(a, 2), (b, 3), (d, 1)} f = {(a, 2), (b, 3), (b, 3), (d, 1)}

f = {(a, 2), (b, 3), (c, 3), (d, 1)}

A = {a, b, c, d}X = {1, 2, 3, 4} The function f:A → X is defined by the arrow diagram below. (a,1)(b,3)(c,4)(d,2) Select the set of pairs that defines a function that is equal to f.

f = {(a, 2), (b, 3), (c, 4), (d, 2)}

Select the function that does not have a well-defined inverse. f:Z→Z f(x)=⌈x+2⌉ f:R→R f(x)=3x+4 f:R→Z f(x)=⌈x⌉ f:R→R f(x)=-2x+5

f:R→Z f(x)=⌈x⌉

A and B are finite sets. The function f:A→B is a bijection. Select the true statement. f∘f-1=IB f∘f-1=IA f∘f-1=f f may not have a well-defined inverse.

f∘f-1=IB

Select the set that corresponds to the relation given in the arrow diagram below. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right. { (1, 2), (2, 2), (2, 4), (3, 3) } { (2, 1), (2, 2), (3, 3), (3, 4) } { (1, 2), (2, 3), (2, 4), (3, 3) } { (2, 1), (2, 2), (3, 3), (4, 2) }

{ (1, 2), (2, 2), (2, 4), (3, 3) }

Which relation on the set {1, 2, 3, 4} is a partial order? { (1, 2), (2, 3), (1, 3), (3, 4) } { (1, 2), (2, 3), (1, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4, 4) } { (1, 2), (2, 3), (1, 3), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4) } { (1, 2), (2, 3), (1, 3), (4, 3) }

{ (1, 2), (2, 3), (1, 3), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4) }

Select the set that corresponds to the relation given in the arrow diagram below: { (1, B), (3, A), (3, D), (4, B), (4, D) } { (A, 3), (B, 1), (B, 2), (D, 3), (D, 4) } { (1, B), (2, B), (3, A), (3, D), (4, D) } { (A, 3), (B, 1), (B, 4), (D, 3), (D, 4) }

{ (A, 3), (B, 1), (B, 4), (D, 3), (D, 4) }

Select the collection of sets that forms a partition of: {1, 2, 3, 4, 5, 6, 7, 8} {1, 2, 5, 7} {3, 4} {8} {1, 2, 5, 7} {3, 4, 6} {8} {1, 2, 5, 7} {3, 4, 6, 8} {2, 4} {0, 1, 2, 5, 7} {3, 4, 6, 8}

{1, 2, 5, 7} {3, 4, 6} {8}

A = {1, 2, {3, 4}, {5, 6, 7}} Select the statement that is true. {3,4}⊆A {3}∈A {1,2}⊆A {1,2}∈A

{1,2}⊆A

For i∈Z+, Ai is defined to be the set of all integer multiples of i. Select the set corresponding to (∩�=24��)∩{x∈Z:1≤x≤30}. {12, 24} {24} {6, 12, 18, 24, 30} ∅

{12, 24}

B={x∈Z:x is a prime number} C={3,5,9,12,15,16} The universal set U is the set of all integers. Select the set corresponding to B∩C. {3, 5, 9, 15} {9, 12, 16} {3, 5} {9, 12, 15, 16}

{3, 5}

C={3, 5, 9, 12, 15, 16} D={5, 7, 8, 12, 13, 15} Select the set corresponding to C⊕D. {3, 9, 16} {3, 7, 8, 9, 13, 16} {3, 5, 7, 8, 9, 12, 13, 15, 16} {5, 12, 15}

{3, 7, 8, 9, 13, 16}

A={x∈Z:x is even} C={3, 5, 9, 12, 15, 16} D={5, 7, 8, 12, 13, 15} The universal set U is the set of all integers. Select the set corresponding to C-(A∪D). {8, 12} {3, 9} {5, 7, 13, 15} {5, 12, 15}

{3, 9}

A={x∈Z:x is even}C={3, 5, 9, 12, 15, 16}D={5, 7, 8, 12, 13, 15} The universal set U is the set of all integers. Select the set corresponding to C-(A∪D). {5, 7, 13, 15} {8, 12} {3, 9} {5, 12, 15}

{3, 9}

A = {x ∈ Z: x is a prime number} B = {4, 7, 9, 11, 13, 14} Select the set corresponding to A∩B. {4, 7, 9, 11, 13, 14} {7, 11, 13} {7, 9, 11, 13} ∅

{7, 11, 13}

A = {x ∈ Z: x is a prime number} B = {4, 7, 9, 11, 13, 14} Select the set corresponding to A∩B. ∅ {7, 9, 11, 13} {7, 11, 13} {4, 7, 9, 11, 13, 14}

{7, 11, 13}

Select the collection of sets that forms a partition of R. {x∈R:x< 2} {x∈R:2< x< 4} {x∈R:4≤x} {x∈R:x≤2} {x∈R:2≤x< 4 }{x∈R:4≤x} {x∈R:x< 4} {x∈R:2≤x≤4} {x∈R:2< x} {x∈R:x< 2} {x∈R:2≤x< 4} {x∈R:4≤x}

{x∈R:x< 2} {x∈R:2≤x< 4} {x∈R:4≤x}

Select the collection of sets that forms a partition of R. {x∈R:x< 4 }{x∈R:2≤x≤4} {x∈R:2< x} {x∈R:x< 2} {x∈R:2≤x< 4} {x∈R:4≤x} {x∈R:x< 2} {x∈R:2< x< 4} {x∈R:4≤x} {x∈R:x≤2} {x∈R:2≤x< 4} {x∈R:4≤x}

{x∈R:x< 2} {x∈R:2≤x< 4} {x∈R:4≤x}

Select the set that is equal to: {3, 5, 7, 9, 11, 13} {x∈Z:3<x<14} {x∈Z:x is prime and 3≤x<14} {x∈R:3≤x<14} {x∈Z:x is odd and 3≤x≤14}

{x∈Z:x is odd and 3≤x≤14}

Select the set that is equal to: {3, 5, 7, 9, 11, 13} {x∈Z:x is prime and 3≤x<14} {x∈Z:x is odd and 3≤x≤14} {x∈Z:3<x<14} {x∈R:3≤x<14}

{x∈Z:x is odd and 3≤x≤14}

Use the definition below to select the statement that is false. A = {x ∈ Z: x is even and 4 < x < 17} 4∉A |A|=7 17∉A 6∈A

|A|=7

Use the definitions below to select the statement that is true. A = {x ∈ Z: x is even} B = {x ∈ Z: −4 < x < 17} ∅⊂B B⊆A A⊂A A is finite.

∅⊂B