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 ...continues

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 intege ...continues

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)

List 16 different relations on the set {0,1} as sets of pairs. State if they are reflexive, transitive, symmetric, antisymmetric.

List 16 different relations on the set {0,1} as sets of pairs. State if they are reflexive, transitive, symmetric, antisymmetric

Discrete Math : Let R be a symmetric relation show that R to the power n is symmetric for all positive integers n.

Let R be a symmetric relation show that R to the power n is symmetric for all positive integers n.

Discrete Math : Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

Relations : A relation R is called circular if a R b and b R c imply that c R a. Show that R is reflexive and circular if and only if it is an equivalence relation.

A relation R is called circular if a R b and b R c imply that c R a. Show that R is reflexive and circular if and only if it is an equivalence relation.

Discrete mathematics

Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}