149) A vertex of a graph is called isolated if
A vertex of degree 0 is called isolated.
150) A vertex of a graph is called pendant if
A vertex is pendant iff it has degree 1.
151) Find the degree of the vertex b in the graph K: =1
152) Find the degree of the vertex a in the graph K: =3
153) Find the degree of the vertex c in the graph K: =5
154) The following graph is called …
The complete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices.
155) Which of the vertices of the following graph is isolated? =there is no such vertex
156) Which of the vertices of the following graph is pendant? =d
157) Find the in-degree of the vertex c in the following graph with directed edges. =3
158) Find the out-degree of the vertex c in the following graph with directed edges. =3
159) Find the in-degree of the vertex d in the following graph with directed edges. =3
160) Find the out-degree of the vertex d in the following graph with directed edges. =0
161) How many subgraphs with at least one vertex does C3 have?
162) How many subgraphs with at least one vertex does C4 have?
163) A simple graph G is called bipartite if …
164. Let G = (V, E) be a graph with directed edges. Then …
165. Which of the following statements is true? A) An undirected graph has an even number of vertices of even degree….
B) An undirected graph has an odd number of vertices of odd degree.
166. The complete graph on n vertices is called
The complete graph on n vertices, denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices.
167. Find the coefficient of the
term in the expansion of
.
C(5,3)*(2)^3
168. Which is the correct expansion of
?
A^4+4*A^3*(2B)^1+6*A^2*(2B)^2+4*A^1*(2B)^3+(2B)^4
169. What is the sum of the coefficients in the
expansion of
?
32
170. What is the sum of the coefficients in the
expansion of
?
0
PS : способ нахождения суммы коэффициентов, если принять а=1 и в=1
(1+1)^5=32
(1-1)^5=0
(a+b)^3=a^3+3*a^2*b+3a*b^2+b^3=8 (1+1)^3=8
(a+2b)^2=a^2+4b+4b^2=9 (1+1*2)^2=9
(a-3b)^2=a^2-6b+9b^2=4 (1-3*1)^2=4
171. Consider the complete graph on 4 vertices. Which of the following statements is true?
…It has 30 edges and each of its vertices has degree 6 …
172. Consider the complete graph on 5 vertices. Which of the following statements is true?
…It has 42 edges and each of its vertices has degree 7 …
173. Try to draw a graph with the following degrees for its five vertices: 1, 1, 1, 2, 3? Which statement is true?
..Such a graph has five edges…
174. Try to draw a graph with the following degrees for its four vertices: 2, 2, 2, 2? Which statement is true?
175. Try to draw a graph with the following degrees for its five vertices: 3, 3, 3, 3, 3? Which statement is true?
176. How many permutations of {1, 2, 3, 4, 5} start with 1? 4!
177. Let S = {1, 2, 3, 4, 5}. How many 3-permutations of S are there? P(5,3)
178. Let A = {a, b, c, d, e, f}. How many 2-combinations of A are there? C(6,2)
179. Let
for i = 1, 2, 3, … Find
{2…7}
180. Let
for i = 1, 2, 3, … Find
{5…+infinity}
181. What rule of inference is used in the following argument: Alice is a runner. Therefore, Alice is either a runner or a cyclist.
182. What rule of inference is used in the following argument: If Alex sleeps all day, he does not do his homework. If he does not do his homework, he will receive F at the school. Therefore, if he sleeps all day, he will receive F at the school.
183. What rule of inference is used in the following argument: It is either rainy today or I play football. It is sunny today. Therefore, I play football.
184. For the following set of premises, what relevant conclusion or conclusions can be drawn?
Every student of this group does not a personal computer. Mary has a personal computer. Ivan does not have a personal computer.
185. For the following set of premises, what relevant conclusion or conclusions can be drawn?
He is either a skier or a runner. He is not a runner. If he is a runner, he will take part at a competition on running.
186. For the following set of premises, what relevant conclusion or conclusions can be drawn?
All students of your group like to prepare a dinner. Alice is a student of your group. David is not student of your group. Mary and Ivan do not like to prepare a dinner.
187. Mathematical induction is a technique for proving that P(n) is true for every positive integer n as follows:
188. How many strings of seven English letters are there those start with X and contain at least two vowels, if letters can be repeated?
26^6-21^6-21^5*5*6
189.
How many solutions are there to the equation
where
and
are nonnegative integers such that
,
and
?
n=4
-(x1<=3 AND x2<3)=(x1>=4 OR x2>=3)
r=15-(4+3+6)=2
C(4+2-1,2)=C(5,2)=10
r=15-(4+6+1)=4
C(4+4-1,4)=C(7,4)=35
r=15-(3+2+6)=4
C(4+4-1,4)=C(7,4)=35
35+35-10=60
190.
How many solutions are there to the equation
where
is
a nonnegative integer such that
,
,
,
,
and
?
n=6
r=24-(1+2+3+4+4+3)=7
C(6+7-1,7)=C(12,7)
191.
How many solutions are there to the equation
where
and
are nonnegative integers such that
and
?
n=4
r=15-8=7
C(4+7-1,7)=C(10,7)
192. A bagel shop has onion bagels, poppy seed bagels, egg bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are there to choose a dozen bagels with at least two of each kind?
n=7
r=12
answer=0 : 7*2=14 but sum=12 and 12<14
193. Find all upper bounds of {4, 18} in the poset ({1, 2, 3, 4, 6, 9, 12, 18, 24, 36, 48, 60}, |).
Числа во множестве должно делиться на 4 и 18 это числа: 36
194. Find the least upper bound of {4, 18}, if it exists, in the poset ({1, 2, 4, 6, 9, 18, 27, 36, 48, 60}, |).
195. Find all lower bounds of {36, 64} in the poset ({4, 6, 9, 16, 24, 36, 48, 64, 76}, |).
Числа 36 и 64 должны делиться на числа во множестве, эти числа: 4
196. Find the greatest lower bound of {36, 64}, if it exists, in the poset ({2, 4, 6, 9, 16, 18, 27, 36, 54, 64, 80}, |). =4
197. Find the greatest lower bound of {18, 48}, if it exists, in the poset ({1, 2, 4, 5, 9, 12, 18, 24, 40, 48, 100}, |). =2
198. Find the least upper bound of {9, 24}, if it exists, in the poset ({1, 3, 6, 9, 10, 18, 24, 36, 40, 66, 98}, |).
Числа во множестве должно делиться на 9 и 24 это числа: нет
199. Determine which of the following posets is a lattice: ({1, 4, 6, 12, 20}, |) … ({3, 5, 10, 50, 150, 300}, |)…
Оба не латис
200. Determine which of the following posets is a lattice: ({1, 3, 6, 9, 18, 36}, |) … ({2, 4, 6, 8, 12, 16, 24, 36, 44, 124}, |)
первое латис, второе нет