In this week of CSC165, I learned
about transitivity, the importance of the order of quantifiers in a statement,
and the guideline to proofs. However,
the majority of the focus during this week of CSC165 was on the assignment that
was due at the end of the week. For this
assignment, I worked with 2 other students in a group of three to produce the
best answers that we could come up with.
Our method of ensuring that we all share the work load equally was by
completing the assignment individually, and then comparing and merging our
ideas and answers to produce a final version.
I felt that of the 5 questions in the assignment, the difficulty level
of the questions increased from question one to questions five. The first question was fairly basic
translation of sentences to its symbolic form, and then negating the symbolic
statements. This section was mostly
applying the correct negation methods and understanding the sentences properly,
which did not cause me a lot of trouble.
For question 2, I realised that I
could substitute some of the predicates such as honor and persistent into one,
which simplifies the statement and made comparing it to the original much
easier. It was the method we agreed to
use to explain whether the statement is the converse, contrapositive, or the
negation.
The third question became a bit
trickier, as it involved creating sentences myself to match the description of
the questions. My plan was to simply
translate the sentences to its symbolic counter parts and vice versa, but soon I
encountered statements that required complicated construction of quantifiers, predicates,
and relationships. The most difficult
question in part 3 was to express that Zorn was enrolled in all courses but Zukes. I spent a lot of time thinking about how to
create the expression ‘there exists only one’ symbolically, and my initial
answer was to say that for all x of classes that Zorn is enrolled in, there
exists a set y of class that Zorn is not enrolled in, if x is in the set of
classes that Zorn is enrolled in, then x does not equal y, and y is equal to
Zukes. At first I thought this answer
was sufficient, however later I discovered that I merely said that Zorn in not
enrolled in Zukes, meaning I had failed to express that Zorn is enrolled in all
other classes. After discussing with my
partners, they showed me that if I said for all x, if x is a class and x is not
equal to Zukes, then Zorn is enrolled in x.
This statement expressed that Zorn is enrolled in all x that is not the
course Zukes.
Question 4 was short, but
although I knew that there are differences between conjunctions and
implications, I had difficulty coming up with exact examples to prove one wrong
and the other right. However after
thinking back to the rules about proving and falsifying universal and
existential quantifiers, I remembered that to prove a universal wrong, all I needed
was to show a counterexample, and to prove an existential wrong, I needed to
show that there are no examples. The
final answers and the explanations are too long for me to put into this blog,
but the main idea is that for implications to be wrong, P must be true and Q
must be false. However for conjunctions
to be wrong, either one being false will do the trick.
For question 5, my way of finding
which implies which was to create truth tables.
I viewed each statement as a predicate, and then created a truth table
to see if the second statement is true or not depending on the antecedent statement. This question took me the longest to solve
however because I did not know exactly how I should be finding out which
statement implies which.
All in all, I know that I have
answered the questions and shared my reasoning with my partners as best as I can,
and I am confident with the results of this assignment.
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