I've solved 2 challenging logic problems today.
As I announced last midnight, I went on my study on discrete mathematics today. I made up my mind to do as many exercises as I can, and so I did. Open my exercise book, you'll see the book is used about 60%, most of which are used today. There are totally 61 problems in the exercise part of Chapter 1.1, and now I've sovle 39 of them. In the problems I solved today, two are marked as challenging. In fact, they are not only challenging but also interesting. I am going to type down the problem and my solutions.
*38. Is the assertion "This statement is false" a propostion?
Solution: According to the definition, a proposition is a declarative sentence that is either true or false, but not both. Analyzing the assertion in the problem, I find that the assertion never make sense no matter I consider it to be true of false. For it's impossible to tell what the truth value of the assertion is, I think it is not a proposition.
*39. The nth statement in a list of 100 statements is "Exactly n of the statements in this list are false."
a) What conclusions can you draw from these statements?
Solution: Since the number of false statements in the list is exclusive, there is only one statement in the list telling the right number. Consequently, there is only one statement is true, namely, there are 99 false statements, which is said in the 99th statement.
Conclution: In the list, only the 99th statement is true, while others are false.
b) Answer part (a) if the nth statement is "At least n of the statements in this list are false"
Solution: Having analyzed the statements, we can find that the nth statement is sufficient to the n-1 statements before it. So if the nth statement is true, the n statements in the front of the list will consequently be true. The nth statement says that at least n of the statements in the list are false, namely, at most 100-n of the statements in the list are true. The maximum of n should be a number that makes n not greater than 100-n. By calculating, we can easily find the maximum of n to be 50.
Conclution: In the list, the 50 statements in the front is true, while others are false.
c) Answer part (b) assuming that the list contains 99 statements.
Solution: If the length of the list is cut down to 99, then the maximum of true statements will be 99-n. Now our task is finding a possible maximum integer that fits the expression n <= 99-n. Easily we can figure out that the integer is 49.
Conclution: In the list, the 49 statements in the front is true, while others are false.
After referring to the key of the problem 39, I found that part (a) and part (b) was solved correctly, but part (c) was not. To key to part (c) is "This cannot happen; it is a paradox, showing that these cannot be statements". Up to now, I can't understand the key to part (c). Why I was wrong and why it should be like that? I think I have to mark the problem down and try to ask for help from google or the website for the book.
*38. Is the assertion "This statement is false" a propostion?
Solution: According to the definition, a proposition is a declarative sentence that is either true or false, but not both. Analyzing the assertion in the problem, I find that the assertion never make sense no matter I consider it to be true of false. For it's impossible to tell what the truth value of the assertion is, I think it is not a proposition.
*39. The nth statement in a list of 100 statements is "Exactly n of the statements in this list are false."
a) What conclusions can you draw from these statements?
Solution: Since the number of false statements in the list is exclusive, there is only one statement in the list telling the right number. Consequently, there is only one statement is true, namely, there are 99 false statements, which is said in the 99th statement.
Conclution: In the list, only the 99th statement is true, while others are false.
b) Answer part (a) if the nth statement is "At least n of the statements in this list are false"
Solution: Having analyzed the statements, we can find that the nth statement is sufficient to the n-1 statements before it. So if the nth statement is true, the n statements in the front of the list will consequently be true. The nth statement says that at least n of the statements in the list are false, namely, at most 100-n of the statements in the list are true. The maximum of n should be a number that makes n not greater than 100-n. By calculating, we can easily find the maximum of n to be 50.
Conclution: In the list, the 50 statements in the front is true, while others are false.
c) Answer part (b) assuming that the list contains 99 statements.
Solution: If the length of the list is cut down to 99, then the maximum of true statements will be 99-n. Now our task is finding a possible maximum integer that fits the expression n <= 99-n. Easily we can figure out that the integer is 49.
Conclution: In the list, the 49 statements in the front is true, while others are false.
After referring to the key of the problem 39, I found that part (a) and part (b) was solved correctly, but part (c) was not. To key to part (c) is "This cannot happen; it is a paradox, showing that these cannot be statements". Up to now, I can't understand the key to part (c). Why I was wrong and why it should be like that? I think I have to mark the problem down and try to ask for help from google or the website for the book.
