09ma6cHW3

09ma6cHW3 - G . Assume that G is a graph with innitely many...

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Ma/CS 6c Assignment #3 Due Wednesday, April 21 at 1 p.m. (35%) 1. (i) Prove that {⇔ , ∧} , {⇔ , ∨} are not complete. (ii) Prove that | , are the only complete binary connectives. (iii) Prove that, “If ... , then ... , else ... ” is not complete, but if we add to it the constants 0,1 it becomes complete. (35%) 2. Let G be a graph with set of vertices V . A coloring of G with k colors ( k = 1 , 2 ,... ) is a map c : V → { 1 , 2 ,...,k } so that if x,y V are adjacent, then c ( x ) 6 = c ( y ). By a finite subgraph of G we mean a graph consisting of finitely many vertices of G , with two such vertices adjacent iff they are adjacent in
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Unformatted text preview: G . Assume that G is a graph with innitely many vertices V = { x 1 ,x 2 ,x 3 ,... } . Show that if every nite subgraph of G has a coloring with k colors, then G has a coloring with k colors. (30%) 3 * . Let { A 1 ,A 2 ,A 3 ,... } be an innite set of formulas in propositional logic. Assume that for every valuation v there is some n (depending on v ) such that v ( A n ) = 1. Show then that there is some xed m with A 1 A 2 A m a tautology. 1...
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