Several Problems

(See attached file for full problem description with proper symbols) --- 2. Let f(x) = x2 +1 and g(x) = {x+1, x> =3; x-1, x<3 so both f and g map R into Find the formula for a. (f+g)(x) b. (f .g)(x) c. (f o g)(x) d. (g o f)(x) 3. Let A = {a,b,c,d} and B = {1,2,3} and let f : A  B be a function . Let g : Z  2Z, where 2Z = {0,+-2,+-4,+-6 …} a. Could f be one to one? Must f be one to one? Explain b. Could f be onto? Must f be onto? Explain c. Could g be one to one? Must g be one to one? Explain d. Could g be onto? Must g be onto? Explain 4. Let ≡ be the relation on Z given by n ≡ m mod 5 iff 5|(n-m). Show that equivalence mod 5 is an equivalen... click for more

Depth-First Search ( DFS ), Undirected Graph, Spanning Tree, Joining Nodes and Ancestors

Let G be an undirected graph, and let T be the spanning tree genereted by a depth-first search of G. Prove that an edge of G that has no corresponding edge in T cannot join nodes in differect branches of the tree, but must necessarily join some node v to one of its ancestors in T.

Discrete mathematics questions

(See attached file for full problem description with proper symbols and equations) --- 1)Prove that for any non-empty sets A x (B-C) = (AxB)-(AxC) 2) Let a,b be integers and m a positive integer. Prove that: ab = [(a mod m ) * (b mod m) mod m ] 3)Prove or disprove (a mod m) + (b mod m) = (a+b) mod m for all integers a and b whenever m is a positive integer. 4) prove that floor(n/2) * ceiling(n/2) = floor (n2/4) 5) For any integer n show that 7n+1 and 15n+2 are relatively prime 6) By induction show that 1*2*3 + 2*3*4 +…n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4 ---

Discrete mathematics

Proper walk through of following proofs required ( for a better understanding ) --- 1) Prove that if n is an odd integer then n2 = 1 mod 8 2) Prove that 5n+3 is divisible by 4 for all integers n>=0

Discrete Math : Equivalences and DeMorgan's Law

Use equivalences to show: ~B ^(A -> B) = ~(B v A) (I used ~ to mean "not") 1. B ^(A -> B) = (B v A) 2. (A v B) -> C = (A -> C) ^ (B -> C) 3. A -> (B v C) = (A -> B) v (A -> C)