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1 .( 10 pts ) For each the following groups of sets , determine whether they form a partition for the set of integers . Explain your answer .
2 .( 10 pts ) Define f : Z→Z by the rule f ( x ) = 6x + 1 , for all integers x .
3 .( 10 pts ) f : R→R and g : R→R are defined by the rules :
4 . ( 10 pts ) Determine whether the following binary relations are reflexive , symmetric , antisymmetric and transitive : For each of the below , indicate whether it is an equivalence relation or a partial order . If it is a partial order , indicate whether it is a total order . If it is an equivalence relation , describe its equivalence classes .
5 .( 10 pts ) Determine whether the following pair of statements are logically equivalent . Justify your answer using a truth table .
6 .( 10pts ) Prove or disprove the following statement :
7 . ( 10 pts ) Prove the following by induction : ∑ _( i = 1 ) ^n▒3i-2 =( 3n^2-n )/ 2
8 .( 10 pts ) Use the permutation formula to calculate the number permutations of the set { V , W , X , Y , Z } taken three at a time . Also list these permutations .
9 .( 10 pts ) Translate the following English sentences into statements of predicate calculus that contain double quantifiers and explain whether it is a true statement .
10 . ( 10pts ) Consider the following graph :