402/502: CSCI Introduction to Theory of Computation (Fall 2007) Instructor: Pranava K. Jha Preliminary Test Sep. 6, 2007 (Thursday) Maximum points: 60
Name:
Q. 1. [10 points] Present the converse and contrapositive of the following statement: If an integer n is divisible by 3, then sum of the digits of n is divisible by 3. Converse:
Contrapositive:
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CSCI 402/502 (Fall 2007): Preliminary Test (Instructor: Pranava K. Jha)
Q. 2. [15 points] The following directed graph corresponds to a binary relation R on the set S = {a, b, c, d, e}.
a
b
c
d
e
Is R reflexive? Explain. Is R symmetric? Explain. Is R transitive? Explain. Q. 3. [10 points] There is something wrong with the following "claim" and "proof." its Detect the error. "Claim": 1 1 1 3 1 + + ... + = - . 1 2 2 3 (n - 1) n 2 n
"Proof": (by induction on n) Basis: n = 1. It is clear that
1 3 1 = - . 1 2 2 n
Induction step: Assuming that the claim is true for n, consider the case for n + 1.
1 1 1 1 1 3 1 + + ... + + = - + 1 2 2 3 (n - 1) n n (n + 1) 2 n n (n + 1) = 3 1 1 1 3 1 . QED - + - = - 2 n n n +1 2 n +1
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