For integers 1<=a<=b<=c, construct a graph G with k(G)=a, lamda(G)=b and delta(G)=c.

ferdysy9
2022-08-08
Answered

For integers 1<=a<=b<=c, construct a graph G with k(G)=a, lamda(G)=b and delta(G)=c.

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Emely English

Answered 2022-08-09
Author has **16** answers

The minimum value of a, b or call can be 1.

k(G)=a(>=1). Thus the graph is complete.

Delta(G) =c(>=1) this mean that a vertex can have at maximum of atleast 1 connection but this is not possible. Also lambda(G) =b(>=1).this means that a vertex is connected by already one other vertex.

Thus we can see that in the graph G, the minimum number of connection is atleast 1 and maximum is atleast 2.

Also,k(G)=a=n-2

Thus,1<=n-2

Or,n>= 3

k(G)=a(>=1). Thus the graph is complete.

Delta(G) =c(>=1) this mean that a vertex can have at maximum of atleast 1 connection but this is not possible. Also lambda(G) =b(>=1).this means that a vertex is connected by already one other vertex.

Thus we can see that in the graph G, the minimum number of connection is atleast 1 and maximum is atleast 2.

Also,k(G)=a=n-2

Thus,1<=n-2

Or,n>= 3

asked 2022-07-07

I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)

asked 2022-04-30

This may be a dumb question.

The Poisson postulates are:

1. $P(n=1,h)=\lambda h+o(h)$

2. $\sum _{i=2}^{\mathrm{\infty}}P(n=i,h)=o(h)$

3. Events in nonoverlapping intervals are independent

What ensures that $\lambda h\in [0,1]$ irrespective of the value of $\lambda $ ?

The Poisson postulates are:

1. $P(n=1,h)=\lambda h+o(h)$

2. $\sum _{i=2}^{\mathrm{\infty}}P(n=i,h)=o(h)$

3. Events in nonoverlapping intervals are independent

What ensures that $\lambda h\in [0,1]$ irrespective of the value of $\lambda $ ?

asked 2022-06-25

Lorenz, Every single line through a point within an angle will meet at least one side of the angle.

I know I have to Show that the parallel postulate 5 implies lorenz, and then lorenz implies parallel postulate 5.

Assume postulate 5 . So we are given AB and a point C not on AB. Choose B on AB draw CD to construct angle ECD= angle BDC.

I just don't get what Lorenz postulate means. Thats where I am getting stuck.

I know I have to Show that the parallel postulate 5 implies lorenz, and then lorenz implies parallel postulate 5.

Assume postulate 5 . So we are given AB and a point C not on AB. Choose B on AB draw CD to construct angle ECD= angle BDC.

I just don't get what Lorenz postulate means. Thats where I am getting stuck.

asked 2022-07-05

So I'm reading about the history of hyperbolic geometry and something like this came up: "two thousand years later, people gave up on trying to derive the fifth postulate from the other 4 and begun studying the consequences of the mathematical structure without the fifth postulate. The result is a coherent theory". My question: were there any possibility of incoherence? I understand some postulates are actually definition of objects, then, clearly, other postulates might be dependent on this other postulate, but it doesn't seem to be the case.

asked 2022-07-23

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asked 2022-06-29

I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook.

Axiom 1.2.1 (Peano Postulates). There exists a set $\mathbb{N}$ with an element $1\in \mathbb{N}$ and a function $s:\mathbb{N}\to \mathbb{N}$ that satisfies the following three properties.

a. There is no $n\in \mathbb{N}$ such that $s(n)=1$.

b. The function s is injective.

c. Let $G\subseteq \mathbb{N}$ be a set. Suppose that $1\in G$, and that $g\in G\Rightarrow s(g)\in G$. Then $G=\mathbb{N}$.

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers $\{1,3,5,7,\dots \}$, or the powers of 5 $\{1,5,25,625\dots \}$, could be constructed (with a different $s(n)$, of course, since $s(n)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?

Axiom 1.2.1 (Peano Postulates). There exists a set $\mathbb{N}$ with an element $1\in \mathbb{N}$ and a function $s:\mathbb{N}\to \mathbb{N}$ that satisfies the following three properties.

a. There is no $n\in \mathbb{N}$ such that $s(n)=1$.

b. The function s is injective.

c. Let $G\subseteq \mathbb{N}$ be a set. Suppose that $1\in G$, and that $g\in G\Rightarrow s(g)\in G$. Then $G=\mathbb{N}$.

Definition 1.2.2. The set of natural numbers, denoted $\mathbb{N}$, is the set the existence of which is given in the Peano Postulates.

My question is: From my understanding of the postulates, we could construct an infinite set which satisfies the three properties. For example, the odd numbers $\{1,3,5,7,\dots \}$, or the powers of 5 $\{1,5,25,625\dots \}$, could be constructed (with a different $s(n)$, of course, since $s(n)$ is not defined in the postulates anyway). How do these properties uniquely identify the set of the natural numbers?

asked 2022-07-07

In Peano arithmetic addition is usually defined with the following two postulates:

$(1a):p+0=p$

$(2a):p+S(q)=S(p+q)$

Lets say I put the successor term of the second postulate on the left? Namely:

$(1b):p+0=p$

$(2b):S(p)+q=S(p+q)$

Are these two definitions of addition equivalent?

$(1a):p+0=p$

$(2a):p+S(q)=S(p+q)$

Lets say I put the successor term of the second postulate on the left? Namely:

$(1b):p+0=p$

$(2b):S(p)+q=S(p+q)$

Are these two definitions of addition equivalent?