# C give a recursive definition of the set s of

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(c) Give a recursive definition of the set S of positive integers that are multiples of 4. Basis Step: 4 ε S Induction Step: ( x)( x ε S x+4 ε S ) or alternatively ( x)( y)( (x ε S y ε S ) x+y ε S ) 4. (10 pts.) Let {a n } be defined by the formula a n = 3n + 2 for n = 1,2,3, .... Define the sequence {b n } recursively by b 1 = 5 and b n+1 = b n + 3 for n = 1,2,3, .... Give a proof by induction that a n = b n for n = 1,2,3, .... Proof: Basis Step: Evidently, a 1 = 3(1) + 2 = 5 = b 1 . Induction Step: To show ( n ε + )( a n = b n a n+1 = b n+1 ), assume a n = b n for an arbitrary n ε + . Then using the induction hypothesis, a n = b n a n + 3 = b n + 3 [Algebra; Motivation: Use the recursive definition of {b n }.] (3n + 2) + 3 = b n+1 [Recursive definition of {b n }, definition of {a n }] 3(n+1) + 2 = b n+1 [Algebra; Motivation: Use the definition of {a n }] a n+1 = b n+1 [The definition of {a n }] Since we have shown that a n = b n implies a n+1 = b n+1 for an arbitrary n ε + , universal generalization implies the truth of ( n ε + )( a n = b n a n+1 = b n+1 ). Finally, since we have verified the hypotheses of the Principle of Mathematical induction, using modus ponens, we may conclude that the proposition ( n ε + )( a n = b n ) is true.

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Brief Answers TEST-2B/MAD2104 Page 2 of 2 5. (15 pts.)
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