. Prove each of the following statements. For each proof, state which of the proof types you have used. (a) If n is an integer and n2 + 5 is odd, then n is even. (b) √ is irrational. Determine whether each of the functions from Z to Z is one to one. (i) ( ) (ii) ( ) (b) Let S= {0,-1, 2}. Find ( ) if ( ) ) Determine whether each of these set is finite, countably infinite, or uncountable. Justify your answer (a) ( ) where (b) Real numbers between 2 and 2.2. a) Decrypt these messages using Caesar cipher “HDW GLP VXP”. (b) List five integers that are congruent to 4 modulo 12. (c) Which positive integers less than 30 are relatively prime to 30? ) (a)

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we consider that a_{u}=0, this make the search paths more flexible and will take the form:

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If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint.

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