ds-t2b-a

# Ε s or alternatively x y x ε s y ε s xy ε 4 10

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ε S ) or alternatively ( x)( y)( (x ε S y ε S) x+y ε 4. (10 pts.) Let {a n } be defined by the formula a n =3 n+2f o r 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 n for n = 1,2,3,. ... Proof: Basis Step: Evidently, a 1 ( 1 )+2=5=b 1 . Induction Step: To show ( n ε + )( a n n a n+1 n+1 ), assume a n n for an arbitrary n ε + . Then using the induction hypothesis, a n n a n +3=b n + 3 [Algebra; Motivation: Use the recursive definition of {b n }.] ( 3 n+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 n+1 [The definition of {a n }] Since we have shown that a n n implies a n+1 n+1 for an arbitrary n ε + , universal generalization implies the truth of ( n ε + )( a n n a n+1 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 n ) is true.

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Brief Answers TEST-2B/MAD2104 Page 2 of 2 5. (15 pts.) Label each of the following assertions with "true" or "false". Be sure to write out the entire word.
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ε S or alternatively x y x ε S y ε S xy ε 4 10 pts Let...

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