# Search results

1. ### Use Extreme Value Theorem

Suppose f:[0,1]->R is continuous, f(0)>0, f(1)=0. Prove that there is a X0 in (0,1] such that f(Xo)=0 & f(X) >0 for 0<=X<Xo (there is a smallest point in the interval [0,1] which f attains 0) Since f is continuous, then there exist a sequence Xn converges to X0, and f(Xn) converges to f(Xo)...
2. ### Prove G contains a cycle of length at least k+1

Tell me if I'm not on the right track. Use induction on k. Pick a path P of maximum length, and suppose vertex vi is a vertex on this path, which has degree at least k, with a set of adjacent vertices {w1,w2,…,wj}, the adjacent vertex set must be on the path. The minimum path length of a...
3. ### Prove G contains a cycle of length at least k+1

This is a graph theory related question. Let G be a simple graph with min. degree k, where k>=2. Prove that G contains a cycle of length at least k+1. Am I suppose to use induction to prove G has a path length at least k first, then try to prove that G has a cycle of length at least k+1...
4. ### How to prove a simple graph is 2-connected?

Problem: "Let G be a simple graph on n vertices such that deg(v)>= n/2 for every vertex v in G. Prove that deleting any vertex of G results in a connected graph." Well, I tried to find the min. case. Let k be the min. deg. of vertex in a simple graph, n is number of vertices in G so k =...
5. ### Graph Theory - bipartite related proof

Okay, so by looking at my original assumption P(n-1)=[(n-1)^2/4]=[(n^2-2n+1)/4]=[n^2/4+(1-2n)/4] So now I need to prove that by adding additional one vertex in result of adding additional (2n-1)/4 edges for all n>5. So P(n)=P(n-1)+(2n-1)/4= [n^2/4+(1-2n)/4+(2n-1)/4]=[n^2/4] But how to...
6. ### Graph Theory - bipartite related proof

How to prove that the number of edges in a simple bipartite graph with n vertices is at most n^2/4? Definition of bipartite graph: a graph whose vertex-set can be partitioned into two subsets such that every edge has one endpoint in one part and one endpoint in the other part. I try to...
7. ### Convergence of Sequence Does {An^2} converges => {An} converges? How to prove it?

So choose sequence An^2 = [(-1)^n]^2 and the sequence An^2 converges to 1, but An is NOT convergent (divergent) sequence. Super! Thanks!
8. ### Convergence of Sequence Does {An^2} converges => {An} converges? How to prove it?

Convergence of Sequence "Does {An^2} converges => {An} converges? How to prove it?" Does sequence {An^2} converges implies to sequence {An} converges? True or False. How to prove it? I kinda think it is false, but couldn’t think of any counterexample to directly proof it. So I try to use...
9. ### Set Theory: Prove the set of complex numbers is uncountable

Yeah, my bet! a, b are real numbers, typo... I got this one. Let C be the set of all complex numbers C={a+bi: a, b are real no.} For any real number r can be mapped to a complex no. by r=r+0i, where r=a and is real no., b=0 is also real no. Let R be set of all real numbers R={r+0i: r...
10. ### Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

Yeah, my bet! a, b are real numbers I've constructed a linear function f: (0,1)->(0,2) defined by f(x)=2x such that f(1/2)=1, when x=1/2 (mid point of domain), y=1 (mid point of range) This linear function is certainly bijection, therefore |(0,1)|=|(0,2)| But how to prove...
11. ### Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

How to prove the open intervals (0,1) and (0,2) have the same cardinalities? |(0, 1)| = |(0, 2)| Let a, b be real numbers, where a<b. Prove that |(0, 1)| = |(a, b)| ----------------------------------- |(0,1)| = |R| = c by Theorem ----------------------------------- I know that we...
12. ### Set Theory: Prove the set of complex numbers is uncountable

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...

Thanks!
14. ### Prove set S is countable iff there exists a surjective/injective function

(a) A nonempty set S is countable if and only if there exists surjective function f:N->S (b) A nonempty set S is countable if and only if there exists a injective function g:S->N There are two way proves for both (a) and (b) (a-1) prove if a nonempty set S is countable, then there exists...
15. ### Cardinality Problem: Prove |A| < |N|

Okay, here is what I got so far. There should be two steps that I need to prove to show |S|<|N| step 1) to construct a injective function f:S->N step 2) to prove the function f:S->N is NOT bijection (mainly NOT surjective function) Step 1) I started with trying to contrust a injection f:S->N...
16. ### Cardinality Problem: Prove |A| < |N|

Prove cardinality of every finite nonempty set A is less then cardinality of natural number N |A|<|N| set A is nonempty finite set natural number N is denumerable (infinite countable set) |A|<|N| if there exist a injective (one-to-one) function f: A->N, but NO bijective function, which...