WEEK 7 JOURNAL: THE PROVE STRUCTURE

    This week, we became more familiar with the proof structure and we learned about how to make the statement precise and short. In programming, every statement must be precise, otherwise the program won't work. However, some programs are precise but because the program is too long, the working procedure will take lots of time. Thus, we should make our statement short and precise. In order to do this, we need to find and delete the sufficient but not necessary condition in our statement.

WEEK 6 JOURNAL: PROVE BY CASES

    This week, I learned how to prove a statement by cases. We can divide a statement into several situations, and show that for every situation, this statement is always true. Then, the statement is true. I find using this method can make some problems be easier to solve, because we only need to proof each part of a statement is true, and then the whole statement becomes true. However, the drawback of this method is that when we proof by cases, we must cover all the cases, otherwise the proof won't make sense. For now we have learned several methods to proof a statement is true or false. In some proofs, we need to use several methods to prove, and I think that's the hard point. Because in the sample problems, we only prove each statement by using one single method, and when we need to prove a statement by using several methods, I think it is hard to choose which method I should use and which I should not use.

WEEK 4 JOURNAL: IMPLICATION, DISJUNCTION AND PROOF

  This week we first talked about the bi-implication and its negation. Bi-implication can be written as A <=> B. This is equivalent to (A => B) ∧ (B => A) or we can write this as (¬A ∨ B) ∧ ( ¬B ∨ A). The negation for bi-implication is  (A ∧ ¬B) ∨ (B ∧ ¬A).
  Then we talked about some thing about how to prove a universally-quantified implication is true. If we want to prove A => B, the key to prove this implication is that we need to find the relationship between A and B, in other words, there must be some common traits between A and B.  We can use Venn diagram and truth table to help us find the common traits between A and B, and then we can judge which statement is true. For example, if we find that A ⊆ B on Venn diagram, we can say that A => B.
   When doing assignment 1, I find it is hard to judge when to use the imply symbol and when to use and. Because according to the truth table, "∧" and "=>" are always the same except vacuous truth. In addition, use vacuous truth is a quicker way to find examples or counter-examples.

WEEK 2 JOURNAL: SENTENCES AND SYMBOLS

  In this week's study, we first learned the rule of how to use conjunction, dis-junction and negation to show the logic connection between two statements. Conjunction means both statements are true, while disjunction means at least one statement is true. The use of negation is to negate a statement. When learning these three idioms, I found that there are some similarities between these three idioms and the python language. In python, the "and" means conjunction, and the "or" means the dis-junction. Moreover, the "not" in python means negation. I find it interesting to program in a logic way, because by doing this, the program I wrote can be more tight and precise. Then we learned some arithmetic laws. By learning those things, I find that we can write the relationship between two statements in many different ways, and they all mean the same logic connections.
  During the tutorial, I find that it is hard to judge which is the best way to express the relationship between two functions. Because we have several writing choices and those choices have similar meanings.
  On Friday's class, our professor gives us a question about folding paper. This question asks us to predict the sequence of ups and downs for any number of times we carry out the folding operation. I use cut-and-try method to induce some regularities about folding paper and I solved this problem at last. Cut-and-try method is helpful when you don't know how to solve a problem. You can try several times and see whether there are some similarities between each time's result.

  Hi, I am Michael Zhang. In this week's study, I learned something about logic. Such as implication, converse, and contrapositive. Learning how to prove or falsify a statement is interesting. However, I find it's hard to make our logic both tight and concise. In addition, I find piazza is a good website for us to discuss questions and solve problems. I think I did okay in the tutorial quiz, for the reason that I did some practice before the quiz.