MATH 2200 Learning Celebration #1 Solutions
Thomas Goller
January 29, 2013
Instructions
• Prove or disprove each of the claims on the back of this sheet.
• You have 45 minutes for 6 problems, so roughly 8 minutes per problem.
• Each problem is worth 10 points.
• Write your solutions on the blank paper provided. Don’t turn in this sheet!
• You don’t have to write the statements of the claims, but label your solutions with the
claim numbers.
• Read the claims carefully!
• Check cases of claims and sketch proofs and counterexamples on scratch paper. Turn
in only your final proofs and counterexamples.
• Write your name at the top of the first sheet of your written solutions. Staple the
pages of your written solutions as you turn them in!
1
General Comments:
• Read the instructions!
• You don’t need to write the claims during a learning celebration, but please write
“proof” before each of your proofs! Do not write “proof” before stating a counterexample.
• Please write neatly! Erase cleanly! If your handwriting is sloppy, then at the very least
leave space between problems!
• Read the claims carefully!
• Above all, don’t forget to think about the claims before trying to prove them! Check
some easy cases. Many of you proved false theorems!!
Problems
Claim 1. Let m and n be integers. If mn is even, then m is even and n is even.
This claim is false! Counterexample: Let m = 0 and n = 1. Then mn = 0 is even, but n
is odd.
Comments:
• Any even integer m and odd integer n, or odd integer m and even integer n, gives a
counterexample.
• Give a counterexample by explicitly stating what m and n are, and then briefly summarizing why the conditions of the claim are met but the conclusion is false. Don’t
write something like “Counterexample: If mn = 6, then m = 3 and n = 2 ...”, because
that is a false statement! (m = 1 and n = 6, among other possibilities, would also
yield mn = 6.) The bulk of the counterexample is really the values of m and n, not
the value of mn.
• The contrapositive of the claim is “if m is odd or n is odd, then mn is odd”. Note
that the “and” has become an “or”! (See Exercise 1.4.6 (d) for why this is so.) Some
of you proved that “if m is odd and n is odd, then mn is odd”, but unfortunately this
is not equivalent to the problem!
Claim 2. Let m and n be integers. If m is odd and n is odd, then m + n is even.
Proof. Since m and n are odd, there exist integers j and k such that m = 2j + 1 and
n = 2k + 1. Then m + n = 2(j + k + 1) is even.
Comments:
2
• The contrapositive statement is “if m + n is odd, then m is even or n is even”. This is
an awkward statement to prove, so instead use a direct proof, as I just did!
Claim 3. Let n be an integer. Then n is odd if and only if 4n − 1 is odd.
This claim is false! Counterexample: Let n = 0. Then 4n − 1 = −1 is odd, but n is even.
Comments:
• Any even integer n gives a counterexample.
• Although ( =⇒ ) is actually true, ( ⇐= ) being false makes the “if and only if”
statement false. There’s no need to give a proof of ( =⇒ ).
• Don’t give your counterexample by saying “Let 4n − 1 = 7. Then n = 2” and so
on. The counterexample is a value for n. You should be choosing n, not solving an
equation to get n.
• The contrapositive of ( ⇐= ) is “if n is even, then 4n − 1 is even”. Many of you proved
this ridiculous claim! Please think about what you are doing!
Claim 4. Let r be irrational. Then
r
3
Proof. Assume for contradiction that
b $= 0 such that 3r = ab . Then r =
irrational.
is irrational.
r
is rational. Then there exist integers a and b with
3
3a
is rational, which contradicts the fact that r is
b
Comments:
• To do a proof by contradiction, assume the conclusion is false and try to deduce a
contradiction. In this case, you should assume that “ 3r is irrational” is false. Don’t
instead assume that “r is irrational” is false – you will be proving the wrong claim!
• Explicitly state what the contradiction is!
• Don’t say “Since r is irrational, assume for contradiction that...” The word “since” is
used when one statement implies another, and r being irrational does not imply that
you should assume something for contradiction.
If you chose to interpret the claim as “If r is irrational, then
could use a proof by contraposition, as follows.
r
3
is irrational”, then you
Proof. Proof by contraposition. Since 3r is rational, there exist integers a and b with b $= 0
such that 3r = ab . Then r = 3a
is rational.
b
Comments:
3
• If you choose to do a proof by contraposition, write “Proof by contraposition”, but do
not then write “If 3r is rational, then r is rational.” When you write a statement in a
proof, you are usually asserting that you have shown it to be true, but in this case that
sentence is what you are trying to prove! If you absolutely must state the contrapositive
of the original “if ... then ...” statement, then follow “Proof by contrapositive” with
a sentence like “It suffices to prove that if 3r is rational, then r is rational” or “I will
prove that if 3r is rational, then r is rational”. But please feel free to do as I did above
and just plunge right into the proof of the contrapositive statement without stating it!
Claim 5. Let r and s be real numbers. If r is rational and s is rational, then rs is rational.
Proof. Since r and s are rational, there exist integers a, b, c, and d with b $= 0 and d $= 0
such that r = ab and s = dc . Then rs = ac
is rational.
bd
Comments:
• Read the claim carefully! Some of you thought r or s was assumed to be irrational
rather than rational, which is a drastically different claim.
• You absolutely must not use the variables a and b for both r and s. If you do, then
you are assuming r = ab and s = ab , which in addition to the rationality of r and s also
establishes that r = s. This is an unjustified extra assumption.
• Use “since”, as I did in the first sentence of the proof, to specify the reason why the
integers a, b, c, and d exist.
Claim 6. Let m and n be integers. If m is odd, then mn is odd.
This claim is false! Counterexample: Let m = 1 and n = 0. Then m is odd but mn = 0
is even.
Comments:
• Any odd integer m and even integer n gives a counterexample.
• This problem went well!
4