Which of the following simple graphs does exist?

a simple graph with five vertices of degrees 1, 2, 2, 2, 3.

a simple graph with seven vertices of degrees 0, 1, 1, 2, 3, 4, 6.

a simple graph with four vertices of degrees 0, 1, 1, 4.

a simple graph with six vertices of degrees 1, 2, 3, 4, 5, 6.

a simple graph with three vertices of degrees 0, 1, 2.

An element a of a poset (S, ≤) is called maximal if

there is no b ∈ S such that a < b

there is no b ∈ S such that b < a

a is not a greatest element of (S, ≤).

Which of the following sets is the equivalence class of 3 for congruence modulo 5?

{…, –7, –2, 3, 8, 13, …}

{…, –8, –3, 3, 9, 15, …}

{…, –5, –1, 3, 7, 11, …}

{…, –10, –5, 0, 5, 10, …}

{…, –8, –3, 2, 7, 12, …}

Which of the following relations on the set of all people is an equivalence relation?

{(a, b) | a and b like the same films}

{(a, b) | a and b share a common grandmother}

{(a, b) | a and b speak a common language}

{(a, b) | a and b don’t understand each other}

{(a, b) | a and b have a common friend}

Find the lexicographic ordering of the following 5-tuples: (1, 0, 1, 0, 1), (0, 1, 1, 1, 0), (0, 1, 0, 1, 0), (0, 1, 0, 0, 1), (1, 1, 0, 0, 0).

(0, 1, 0, 0, 1), (0, 1, 0, 1, 0), (0, 1, 1, 1, 0), (1, 0, 1, 0, 1), (1, 1, 0, 0, 0).

(0, 1, 0, 1, 0), (0, 1, 0, 0, 1), (0, 1, 1, 1, 0), (1, 1, 0, 0, 0), (1, 0, 1, 0, 1).

(1, 1, 0, 0, 0), (1, 0, 1, 0, 1), (0, 1, 0, 1, 0), (0, 1, 1, 1, 0), (0, 1, 0, 0, 1).

(1, 0, 1, 0, 1), (0, 1, 1, 1, 0), (0, 1, 0, 1, 0), (0, 1, 0, 0, 1), (1, 1, 0, 0, 0).

(0, 1, 0, 1, 0), (0, 1, 0, 0, 1), (1, 1, 0, 0, 0), (0, 1, 1, 1, 0), (1, 0, 1, 0, 1).

Let S = {1, 2, 3, 4, 5, 6, 7, 8}. Which of the following collections of sets forms a partition of S?

 {1, 4, 8}, {3, 5, 7}, {2, 6}

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

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

Which of the following sets is the equivalence class of 1 for congruence modulo 3?

{…, –5, –2, 1, 4, 7, …}

{…, –6, –3, 0, 3, 6, …}

{…, –4, –1, 2, 5, 8, …}

{…, –7, –4, –1, 1, 4, 7, …}

{…, –9, –5, 1, 5, 9, …}

Find the lexicographic ordering of the following strings of lowercase English letters: computer, computing, comma, competent, computable.

comma, competent, computable, computer, computing

computer, computable, computing, comma, competent

comma, computable, computer, computing, competent

computer, competent, comma, computing, computable

computable, computer, computing, competent, comma

Let S = {0, 1, 2, 3}. With respect to the lexicographic order based on the usual “less than” relation find all pairs in S  S less than (1, 3).

(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2)

(0, 0), (1, 1), (1, 2), (2, 0), (2, 1), (0, 1), (0, 2), (0, 3)

(2, 0), (2, 1), (2, 2), (3, 0), (1, 0), (1, 1), (1, 2), (0, 0), (0, 3)

Which of the following relations on {0, 1, 2, 3, 4} is an equivalence relation?

{(0, 0), (0, 3), (1, 1), (2, 2), (2, 4), (3, 0), (3, 3), (4, 2), (4, 4)}

{(0, 0), (0, 1), (1, 0), (2, 2), (2, 4), (3, 3), (4, 2)}

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

{(0, 0), (1, 1), (1, 3), (2, 2), (3, 2), (3, 3), (3, 4), (4, 4)}

{(0, 0), (0, 1), (1, 2), (0, 2), (2, 1), (3, 4), (4, 1), (3, 1)}

List the ordered pairs in the relation on {1, 2, 3, 4} corresponding to the matrix .... (where the rows and columns correspond to the integers listed in increasing order).

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

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

{(0, 2), (0, 3), (1, 0), (1, 2), (2, 0), (2, 2), (3, 0)}

{(0, 1), (0, 2), (0, 3), (1, 1), (2, 0), (2, 2), (3, 0)}

{(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (4, 2)}

Let R1 = {(a, b) | a = b + 1} and R2 = {(a, b) | a + b ≤ 3} be relations on {0, 1, 2, 3}. Find R1 - R2.

{(0, 0), (0, 2), (1, 2), (2, 0), (3, 0)}

Let R = {(a, b) | a ≤ b} be a relation on the set of integers. The relation R is

reflexive, antisymmetric and transitive

reflexive, symmetric and transitive

non-reflexive and non-transitive

non-antisymmetric and non-symmetric

A relation S on a set B is called antisymmetric if

both (a, b) and (b, a) belong to S only if a = b for all a, b ∈ B

(b, c) ∈ whenever (c, b) ∈ S for all b, c ∈ B

(b, b) ∈ S for every element b ∈ B

whenever (b, c) ∈ S and (c, a) ∈ S then (b, a) ∈ S for all a, b, c ∈ B

A relation S on a set B is reflexive if

(b, b) ∈ S for every element b ∈ B

(b, c) ∈ S whenever (c, b) ∈ S for all b, c ∈ B

whenever (b, c) ∈ S and (c, a) ∈ S then (b, a) ∈ S for all a, b, c ∈ B

it is both symmetric and antisymmetric

List the triples in the relation {(a, b, c) | a, b and c are positive integers with 1 < a + b < c ≤ 4}

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

{(0, 2, 3), (0, 2, 4), (2, 0, 3), (2, 0 , 4), (1, 1, 3), (1, 1, 4), (0, 3, 4), (3, 0, 4)}

{(1, 2, 4), (0, 1, 3), (0, 2, 4), (0, 3, 4)}

{(0, 1, 2), (0,1, 3), (0, 1, 4), (0, 2, 3), (0, 2, 4), (1, 0, 2), (1, 0, 3), (1, 0, 4), (2, 0, 3), (2, 0, 4)}

{(1, 2, 1), (0, 1, 3), (1, 1, 2), (2, 1, 1), (1, 0, 3), (3, 0, 1)}

List the ordered pairs in the relation on {1, 2, 3} corresponding to the matrix ..... (where the rows and columns correspond to the integers listed in increasing order).

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

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

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

{(1, 1), (1, 3), (3, 1), (3, 3)}

Let R = {(a, b), (b, c), (c, a), (d, b)} and S = {(a, a), (b, b), (c, c), (d, a)} be relations on A = {a, b, c, d}. Find S * R

{(a, b), (b, c), (c, a), (d, b)}

{(a, a), (b, b), (c, c), (d, a)}

{(a, a), (b, a), (c, a), (c, b), (d, b), (d, c)}

{(a, a), (a, b), (b, b), (b, c), (c, a), (c, c), (d, a), (d, b)}

{(b, a), (c, b), (a, c), (b, d), (a, d)}

Let R1 = {(1, 1), (1, 2), (2, 2), (2, 3), (3, 3), (3, 4)} and R2 = {(1, 2), (2, 1), (2, 4), (3, 1), (3, 2), (3, 4)} be relations from {1, 2, 3} to {1, 2, 3, 4}. Find R1⋂R2

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

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

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

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

. List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3} where (a, b) ∈ R if and only if a + b = 3.

{(0, 3), (1, 2), (2, 1), (3, 0)}

{(0, 3), (1, 2), (2, 1), (3, 0), (4, –1)}

{(0, 0), (0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}

{(0, 0), (1, 1), (2, 2), (3, 3)}

{(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 4)}

dc1-23

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1st Grade

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MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A vertex of a graph is called pendant if

its degree equals 0

its degree equals 2

its degree equals 1

it is not isolated

there is no true answer

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A vertex of a graph is called isolated if

there is no true answer

its degree equals 0

its degree equals 2

its degree equals 1

it is not pendant

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A subgraph of a graph G = (V, E) is

a graph H = (W, F) where W⋂V≠∅ and F⋂E≠∅

a graph H = (W, F) where W ⊆ V and F ⊆ E

a graph H = (W, F) where V⊂ W and E ⊂ F

a graph H = (W, F) where W ≠ V and F ≠ E

a graph H = (W, F) where W⋂V=∅ and F⋂E=∅

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

The union of two simple graphs G1=(V1,E1) and edge set G2=(V2,E2) is

the undirected graph with vertex set V1 ⋃ V2 and edge set E1 ⋃ E2 such that |E1⋃E2|=|E1|+|E2|.

the undirected graph with vertex set V1 ⋃ V2 such that |V1⋃ V2|=|V1|+|V2| and edge set E1 ⋃ E2

the multigraph with vertex set V1⋃ V2 and edge set E1 ⋃ E2 such that |E1 ⋃ E2|=|E1|+|E2|

the simple graph with vertex set V1 ⋃ V2 and edge set E1 ⋃ E2

the simple graph with vertex set V1 ⋂ V2 and edge set E1 ⋂ E2

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A directed graph differs from a directed multigraph since

every directed multigraph has multiple edges and has no loops

every directed multigraph has multiple edges

every directed graph has multiple edges and has no loops

every directed graph has loops and multiple edges

every directed multigraph has loops and has no multiple edges.

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A pseudograph differs from a multigraph since

every pseudograph has multiple edges and has no loops

every multigraph has loops and has no multiple edges

every multigraph has multiple edges and has no loops

every multigraph has no multiple edges and loops

every pseudograph has no multiple edges and loops

MULTIPLE CHOICE QUESTION

30 sec • 1 pt

A simple graph differs from a multigraph since

every multigraph has loops

every multigraph has multiple edges or loops

every simple graph has loops

every simple has loops or multiple edges

every multigraph has multiple edges

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dc1-23

A vertex of a graph is called pendant if

A vertex of a graph is called isolated if

A subgraph of a graph G = (V, E) is

The union of two simple graphs G1=(V1,E1) and edge set G2=(V2,E2) is

A directed graph differs from a directed multigraph since

A pseudograph differs from a multigraph since

A simple graph differs from a multigraph since

Which of the following simple graphs does exist?

How many edges are there in an undirected graph having 3 vertices each of degree 3 and 5 vertices each of degree 5?

How many edges are there in an undirected graph with 7 vertices each of degree 4?

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