MATH/COMP61-03 Spring 2014
Final Exam Solutions
Every question is worth 10 points.
As stated on the exam, it was required to justify answers, for full credit.
Problem 1. Using a truth table, prove that (p q) (r q) = (p r) q.
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MATH/COMP61-03 Spring 2014 Notes:
January 30th
Andrew Winslow
1
A Nice Conjecture
Last lecture someone in the audience gave the following conjecture:
Conjecture. Let A and B be sets. If A B = B A, then A = B.
I sort of danced around it, some people seemed

MATH/COMP61-03 Spring 2014 Notes:
January 28th
Andrew Winslow
1
Operators on Sets
Just like algebra and boolean algebra, there are operators on sets. Here are some:
Denition. The union of two sets A and B, written A B, is cfw_x : x A or x B.
Denition. The

MATH/COMP61-03 Spring 2014 Notes:
January 23rd
Andrew Winslow
1
Lists
Denition. A list is an ordered collection of objects.
The notation for lists to start with an open parenthesis, then the objects in order
separated by commas, then a closed parenthesis.

MATH/COMP61-03 Spring 2014 Notes:
January 16th
Andrew Winslow
1
Denitions
Denitions are the starting point for all mathematical reasoning. Here is a dention:
Denition. An integer n is even provided n it is divisible by 2.
This dention uses other terms tha

MATH/COMP61-03 Spring 2014 Notes:
January 21st
Andrew Winslow
1
Counterexamples
A statement that hasnt been proved is a conjecture:
n
Conjecture. If n > 4 is an integer, then 22 + 1 is composite.
One goal of mathematics is to convert as many conjectures a