ArsDigita University
Month 2:
Discrete Mathematics  Professor Shai Simonson
Problem Set 1 – Logic, Proofs, and Mathematical Reasoning
1. Logic Proofs.
a. Prove that
a
→
b
is equivalent to
¬
b
→
¬
a
using a truth table.
b. Prove it using
algebraic
identities.
c. Prove that
a
→
b
is not equivalent to
b
→
a
.
2. Aristotle’s Proof that the Square Root of Two is Irrational.
a. Prove the
lemma,
used by Aristotle in his proof, which says that if
n
2
is
even, so is
n.
(Hint:
Remember that
a
→
b
is equivalent to
¬
b
→
¬
a
).
b. Prove that the square root of 3 is irrational using Aristotle’s techniques.
Make sure to prove the appropriate lemma.
c. If we use Aristotle’s technique to
prove
the untrue assertion that the square
root of 4 is irrational, where
exactly
is the hole in the proof?
d. Using the fact that the square root of two is irrational, prove that sin (
π
/4)
is irrational.
3. In ADUball, you can score 11 points for a goal, and 7 for a near miss.
a. Write a Scheme program that prints out the number of goals and the
number of near misses to achieve a given total greater than 60.
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 Spring '09
 Logic, Tn, Euclid, ArsDigita University

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