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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: