This week I learned about conjunctions and disjunctions. I learned a few new logic notation symbols for " and" (used in conjunction statements) and "or" (used in disjunction statements). "And" combines two elements so that both of them are true, and "or" combines statements so at least one of them is true.We then applied both of these to the concept we learned in the previous class, predicates. I found this part of the lecture interesting, as I learned to combine this new topic with a previous one.
I then learned about the negation and its symbol. We did a few examples with it, recognizing that the negation of a universal quantification is an existential quantification, and the negation of an existential quantification is a universal quantification. I found the fact that the negation of both types of quantification is each other very interesting.
We then talked about scope, and how a parentheses helps define the scope of a statement. The placement of the parentheses is very important in determining the order in which the statement if evaluated. This makes all the difference in determining whether statements are true or false.
We then learned about truth tables and how to use them. It turns out that truth tables can be used for a variety of different reasons, including evaluating expressions, determining the satisfiability of a statement, and proving equivalence between statements. Professor Zhang even demonstrated through an example how we could prove DeMorgan's law through truth tables.I found truth tables to be very practical, and will definitely be using them in the future.
We briefly reviewed some other laws, such as the associative law and the commutative law,(among others) before ending the lecture. I will further review these laws before the next lecture in order to refresh my memory.
In order to remember the material seen in this lecture, I will review the course notes and go over the Professor Zhang's slides.
In the tutorial, we reviewed the exercise given beforehand. A lot of people, including myself, had trouble getting through the worksheet. I was slightly more confident about the material after T.A. Elias explained the questions further, but I still had my doubts. I did not feel confident as I did the quiz at the end of the tutorial, but I still managed to think of a solution which made sense. This is what I put on my paper, and I hope it is correct.
Thursday, September 25, 2014
Thursday, September 18, 2014
Week 2
This week in class we learned more about quantifiers.
First, we reviewed the methods to prove and disprove universal and existential quantification. We did a few examples with data tables and Venn diagrams. Personally, I enjoyed the examples with the Venn diagrams because they helped me visualize the universe where the quantifications take place, which parts of this universe are empty, and which ones contain elements. (Professor Zhang included X and O symbols to show which parts of the Venn diagram are empty and which contains elements) We then used logic symbols to solve more problems with Venn diagrams.
Later, we moved on to quantifiers about sets. We used symbols about sets, which I found useful as a review of these aforementioned symbols I learned about in the past.
We then moved on to evaluating quantified claims. I enjoyed examples Professor Zhang included with Python notation (we were shown two boxes with code and we were to choose which one corresponded with the quantification) since they demonstrated how to convert human language into computer language.
Next, I got to learn a new concept, sentences and statements. Professor Zhang demonstrated the fundamental differences between sentences and statements, where sentences have subjects and predicates, and a statement is a sentence.
Then, we talked about predicates. I learned that the predicate can be represented by L(x) (where x has property L.)
The next topic we covered was implications. I found the relationship between the antecedent and the consequent interesting since it is not causal in nature. One does not cause the other. We learned about contrapositives, or the contradiction of the subject and the predicate, and how to express them in logic symbols. We used if-then statements to decide which part of the sentence was the subject and which was the predicate. I found this part of the lecture challenging as well, but I plan to review the course notes and Professor Zhang's slides to better understand this topic.
Next, we talked briefly about what I believed to be the most interesting part of the lecture, vacuous truths, a truth that claims that all members of a group have a particular property. Statements such as "pink elephants have 7 legs" could be true because there are no pink elephants around. I found this interesting because it is a type of truth that could apply to members of a group that don't exist (i.e. pink elephants). Next we talked about equivalence (statements that mean the same thing), its vocabulary (i.e. necessary and sufficient.), and idioms (other ways of saying things in math).
During the tutorials, we solved tutorial exercise number 1. This exercise included some proofs we were to express in natural language for the first questions, and in Venn diagrams in the second question. I found these proof exercise challenging at first, but with some help from T.A. Elias, I was able to solve them. I then took the quiz which was verifying another proof with Venn diagrams and natural language. I think I did fairly well.
First, we reviewed the methods to prove and disprove universal and existential quantification. We did a few examples with data tables and Venn diagrams. Personally, I enjoyed the examples with the Venn diagrams because they helped me visualize the universe where the quantifications take place, which parts of this universe are empty, and which ones contain elements. (Professor Zhang included X and O symbols to show which parts of the Venn diagram are empty and which contains elements) We then used logic symbols to solve more problems with Venn diagrams.
Later, we moved on to quantifiers about sets. We used symbols about sets, which I found useful as a review of these aforementioned symbols I learned about in the past.
We then moved on to evaluating quantified claims. I enjoyed examples Professor Zhang included with Python notation (we were shown two boxes with code and we were to choose which one corresponded with the quantification) since they demonstrated how to convert human language into computer language.
Next, I got to learn a new concept, sentences and statements. Professor Zhang demonstrated the fundamental differences between sentences and statements, where sentences have subjects and predicates, and a statement is a sentence.
Then, we talked about predicates. I learned that the predicate can be represented by L(x) (where x has property L.)
The next topic we covered was implications. I found the relationship between the antecedent and the consequent interesting since it is not causal in nature. One does not cause the other. We learned about contrapositives, or the contradiction of the subject and the predicate, and how to express them in logic symbols. We used if-then statements to decide which part of the sentence was the subject and which was the predicate. I found this part of the lecture challenging as well, but I plan to review the course notes and Professor Zhang's slides to better understand this topic.
Next, we talked briefly about what I believed to be the most interesting part of the lecture, vacuous truths, a truth that claims that all members of a group have a particular property. Statements such as "pink elephants have 7 legs" could be true because there are no pink elephants around. I found this interesting because it is a type of truth that could apply to members of a group that don't exist (i.e. pink elephants). Next we talked about equivalence (statements that mean the same thing), its vocabulary (i.e. necessary and sufficient.), and idioms (other ways of saying things in math).
During the tutorials, we solved tutorial exercise number 1. This exercise included some proofs we were to express in natural language for the first questions, and in Venn diagrams in the second question. I found these proof exercise challenging at first, but with some help from T.A. Elias, I was able to solve them. I then took the quiz which was verifying another proof with Venn diagrams and natural language. I think I did fairly well.
Wednesday, September 17, 2014
Week 1
This week I learned the importance of mathematics for all Computer Science applications. In order to make compelling and interesting programs, mathematics is required. From artificial intelligence, to graphics, mathematical applications are needed (i,e, probability, linear algebra, and geometry).
I also learned about precision and ambiguity in computer languages. I learned that computers are not only extremely precise, but they require a certain type of mathematical precision. (By mathematical precision I mean a certain type of precision where the meanings of the words put into the program are restricted.) Then, I learned about the relationship between human language and computer language, and how they translated into one another. Professor Zhang managed to include both human language and computer language into his examples with python code and Venn diagrams, which helped me understand the relationship between the two aforementioned elements better. I also learned how human language has an adequate level of precision and ambiguity (or intuition) to help us solve problems effectively.
I then learned about more problem solving with quantifiers. First, Professor Zhang gave us useful steps to follow in order to solve problems effectively. These steps include: Understanding the problem, trying more than one plan, executing the plan, and then reviewing the plan by convincing a doubtful peer. I then learned about universal and existential quantification, their respective symbols, and how to prove and disprove each statement (To prove universal quantification: Prove that all elements in the domain satisfy the quantification, to disprove show one counterexample that does not satisfy the quantification. To prove existential quantification show that there is one example that satisfies the quantification, to disprove show that all the elements of the domain are counter examples that do not satisfy the quantification.) We did a few examples with a table of employees that had different salaries to demonstrate both types of quantification and their vocabulary. Vocabulary words that correspond to universal quantification are: all, every, or any word that describes the entirety of a population (ex. male and female.) Vocabulary words that correspond to existential quantification are: Some, there exists, at least one, where not all members of the population correspond to the quantification.
I enjoyed the examples Professor Zhang included in his explanations this week, since not only did they apply to real world situations, but they were also humorous. I particularly enjoyed the problem where two people meet on the street, and one is trying to figure out what the ages of the other person's children are. We solved this particular problem algebraically as a class, assigning values of x,y, and z to each child and using inequalities. This showed me the importance of problem solving with mathematics not only in computer science, but in day-to-day situations. I found this problem compelling, and the humorous aspect of its presentation decreased my anxiety caused by not grasping the entirety of the information being presented.
I initially found using the universal quantification and existential quantification symbols in problems slightly challenging, which led to a slight frustration, but as did examples of a class, I started to grasp the material. Currently, I feel confident about the material covered this week, but in order to remember the information I will review Professor Zhang's slides and the course notes throughout the rest of the week.
There were no tutorials or assignments due.
I also learned about precision and ambiguity in computer languages. I learned that computers are not only extremely precise, but they require a certain type of mathematical precision. (By mathematical precision I mean a certain type of precision where the meanings of the words put into the program are restricted.) Then, I learned about the relationship between human language and computer language, and how they translated into one another. Professor Zhang managed to include both human language and computer language into his examples with python code and Venn diagrams, which helped me understand the relationship between the two aforementioned elements better. I also learned how human language has an adequate level of precision and ambiguity (or intuition) to help us solve problems effectively.
I then learned about more problem solving with quantifiers. First, Professor Zhang gave us useful steps to follow in order to solve problems effectively. These steps include: Understanding the problem, trying more than one plan, executing the plan, and then reviewing the plan by convincing a doubtful peer. I then learned about universal and existential quantification, their respective symbols, and how to prove and disprove each statement (To prove universal quantification: Prove that all elements in the domain satisfy the quantification, to disprove show one counterexample that does not satisfy the quantification. To prove existential quantification show that there is one example that satisfies the quantification, to disprove show that all the elements of the domain are counter examples that do not satisfy the quantification.) We did a few examples with a table of employees that had different salaries to demonstrate both types of quantification and their vocabulary. Vocabulary words that correspond to universal quantification are: all, every, or any word that describes the entirety of a population (ex. male and female.) Vocabulary words that correspond to existential quantification are: Some, there exists, at least one, where not all members of the population correspond to the quantification.
I enjoyed the examples Professor Zhang included in his explanations this week, since not only did they apply to real world situations, but they were also humorous. I particularly enjoyed the problem where two people meet on the street, and one is trying to figure out what the ages of the other person's children are. We solved this particular problem algebraically as a class, assigning values of x,y, and z to each child and using inequalities. This showed me the importance of problem solving with mathematics not only in computer science, but in day-to-day situations. I found this problem compelling, and the humorous aspect of its presentation decreased my anxiety caused by not grasping the entirety of the information being presented.
I initially found using the universal quantification and existential quantification symbols in problems slightly challenging, which led to a slight frustration, but as did examples of a class, I started to grasp the material. Currently, I feel confident about the material covered this week, but in order to remember the information I will review Professor Zhang's slides and the course notes throughout the rest of the week.
There were no tutorials or assignments due.
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