Week 3 Slog

During the third week of CSC165, Professor Heap introduced us to Conjunctions, Disjunctions and Negations. We learnt the different notations used to signify conjunctions and injunctions. Throughout the week, I also improved my ability of transferring english statements to mathematical symbols and have improved overall. In addition to this, we learnt how negation has the ability to change symbols. Hence, we were introduced to DeMorgan's Laws.

We often use conjunctions when combining two statements with an "and". This therefore symbolises that both the original two statements are true. Let us take an example, when making the statement, "There is a pen and telephone" we can therefore use a conjunction since the statement uses the word "and". Disjunctions on the other hand are different from conjunctions since they are joined together in order to prove that either one of them is true. It however does not prove that both are true. When negating a statement we put a "not" operator at the front in order to prove that the opposite of that statement is true. Hence, as an example, not(True) returns the opposite of the boolean value True, hence False. This therefore leads us to discuss DeMorgan's Laws. DeMorgan's Laws prove that when one negates an entire statement, every other element in the statement also turns into a negative statement.

As seen above the not operator is applied to s(1) as well as s(2). In addition, the Disjunction operator also changes to the the Injunction operator. Professor Heap taught us how this can be proven through Sets. This therefore not only increased my understanding of the theory behind this but also allowed me to understand how one can use this.

In order to illustrate implications and symbolic statements, we were introduced to truth tables which really allowed me to gain a better understanding of implications, conjunctions and disjunctions.

Lastly, we were also introduced to some simple terminology namely Tautology, Satisfiability and Unsatisfiability. One makes a judgement that a statement is satisfiable when the premises leads well into the conclusion. However, the opposite of this would mean that the statement is not satisfiable since there is a contradiction if one of our test cases fail. A Tautology is  when one explores various test case values in order to come to a conclusion.

Overall, this week really allowed me to gain a better understanding of using Logic in Computer Science. I believe that truth tables really allowed me to gain a better grasp on the subject material.

Week 1 and 2

In the first week of the course, Professor Heap went over the basics of set theory which are the basic fundamentals of Mathematical reasoning in Computer Science. We were introduced to the method of solving Mathematical problems which is common in both Computer Science and Mathematics. Firstly we must understand the problem at hand and analyse the situation thoroughly. We must be able to familiarise ourselves with any relevant given values and understand which variable we are required to find. In order to make the problem simpler, one must restrict the language and therefore strive for clarity. This therefore, allows us to reach precision in our answer. Next, one must be able to generate an effective plan to tackle the problem at hand. The plan must include ways in which we can use given values in the question to carry out a series of steps to generate an answer. Next, we must carry out the plan in order to find our solution. The final step, allows us to review our answer to match with the accepted answer. If such solution matches, we can conclude that our answer is correct, otherwise we must check our plan to verify there we had gone wrong. We therefore practiced this procedure on numerous problems, many involving Python code. I found this part of the lesson particularly useful as it gave me a headstart into problem solving in Mathematics. It also allowed me to find the relationship of this to Mathematics. Further on this week, we learnt about representing sets in diagrams which allowed me to understand how sets are represented in graphical form. This also appeared in our tutorial exercise which I found fairly straightforward to understand.

In the second week, we learnt about implications, converses and contrapositives. An implication is relates to a cause and effect, or in mathematical terms P(x) -> Q(x). This is mainly found in conditional statements. The opposite of this  is known as the converse, which is mainly known as Q(x) -> P(x). Furthermore, the contrapositive is known as not(P(x)) -> not(Q(x)). I have found this section of the course particularly challenging but hope to reach full understanding of the course material by the end of this week. However, due to the rigour of the material, this part of the course was by far the most enjoyable in CSC165