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Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive.A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement “Fred is happy,” because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement “John is happy,” because John is happy slightly less than half the time. Use these truthvalues to solve below exercise.

The nth statement in a list of 100 statements is “Exactly n of the statements in this list are false.”

  1. What conclusions can you draw from these statements?
  2. Answer part (a) if the nth statement is “At least n of the statements in this list are false.”
  3. Answer part (b) assuming that the list contains 99 statements
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A)The  n^{th} statement in a list of 100 statements is:
"Exactly  n of the statements in this list are false."
1)Exactly  1 of the statements in this list are false.
2)Exactly  2 of the statements in this list are false.
3)Exactly  3 of the statements in this list are false.
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98)Exactly  98 of the statements in this list are false.
99)Exactly  99 of the statements in this list are false.
100)Exactly  100 of the statements in this list are false.
Let 1) is true.
so exactly one of the statement is false.let it be 3).so other 99 statements are true.But this is clearly impossible.
In a similar fashion we can eliminate statements (2) to (98).

Can (98) be true?
Then 98 statements are false.
Which two are true?
One of them is (98); suppose the other is (37).
Then (98) says "Exactly 98 statements are false."
and (37) says "Exactly 37 statements are false."
And we have a contradiction. so (98) cannot be true.

Can (100) be true?
Then all statements are false, including (100).
This leads to a logical dilemma.

The only remaining statement is (99).
It says "Exactly 99 statements are false."
And this is possible!
Statements (1) through (98) and (100) will be false.
B)Same way we can solve it. So we will get 1-50 are true and 51-100 are false. 
C)This cannot happen.

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Answer of question a):
n-th statement: "Exactly n of the statements are false"
So, 1st statement: Exactly 1 of the statements are false
So, 2nd statement: Exactly 2 of the statements are false

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each statement is contradictory to each other.if I say 1st statement is true i.e exactly 1 of the statement is false,so it means out of 100 statements 1 statement is false,correct?so,all 99 are correct statements. If rest 99 are correct, then let say, statement no. 3 which is "exactly 3 of the statements are false" should be correct, but if that statement no. 3 is correct then 3 statements are false,but according to 1st statement that is wrong,similarly u can see that, all 100 statements are contradict within themselves,so maximum one can be true, and thats possible when only statement no. 99 is true, that is "exactly 99 of the statements are false" and yeah thats true and only 1 statement is correct which is statement no. 99. So,all statements no. (1-98) and 100 are wrong,and 99 is only correct.
Answer of question b) Part1:

 if the nth statement is “At least of the statements in this list are false.”

statement 1: At least 1 of the statements in this list are false.

Statement 2: At least 2 of the statements in this list are false.

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So if i say statement 1 is false, then it means that no statement is false,but I am saying that statement 1 is false,its again conflicting,so statement 1 must not be false,it should be true,

Similarly for statement 2,if i say it is false,then it means, no of false statements is<2 i.e 1, but we have already declared that statement 1 must be true,so again statement 2 need to be true.
Same thing happens for statement no.3,4,5....,50, because,

For statement 50, it says:"At least 50 of the statements in this list are false.", I have already declared that 1-49 are true,so if statement no. 50 is false,then no. of false statements<50, but already 49 statements are true and 0 statements are false,so statement no. 50 must also be true.
For statement no 51, if i say it is true then,atleast 51 statements have to be false, but already 50 statements are true and 50 are remaining to be checked,so statement no. 51 must not be true,so it should be false,so if it is false it means no of false statements<51,and its matching with our logic,because already 50 are true,so rest are all false,so 1-50 are true and 51-100 are false.

Answer of question b) Part 2:

In a same manner, what we have seen just above 1-49 is true,

What about statement no. 50?

If it is true, there must be at least 50 false statements. As 1−50 are true in this case, this can never be the case as we only have 99 statements

If it is false, it is not the case that at least 50 of the statements are false. As 1−49 are true, this can only be the case if at least one of the statements in the interval 52−99 is true. This can never be the case for the above reason. Thus statement 51 is neither true or false.
So its a paradox.

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Lets say n=4.

So 4 th statement says that exactly 4 statements are false.

Lets say the above statement is TRUE. Then from the remaining 99 excluding statement 4, we have 4 false statements. So total number of true statements is 96.

Now if statement 4 is FALSE, then number of false statements from remaining statements can range from 0 to 99 but not 3 or [0,99]-{3}.

Part B:

At leat 4 of the statements are false.

If above statement is TRUE then min 4 and max 100 statements can be false.

If the above statement is FALSE then again Total number of false statements can be from 1 to 100. Bcoz the statement itself was wrong and we have no idea about truth value of other statements except current statement.

In similar manner you can understand part C.
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a) Some number of the statements are going to be true. Therefore exactly one of the statements must be true and the remainder are false.

b) The 100th statement is not true because it is asserting that all the statements are false, which therefore it must itself be false. That makes the first statement true. If the 99th statement were true then it could be concluded that the remaining statements are false, which contradicts the truth of statement 99. Therefore statement 99 is false. Therefore statement 2 is true. Therefore 1 through 50 are all true and statements 51 through 100 are all false.

c) If odd numbers are present, then it would be a contradiction when we got to the middle. If
there are only three statements, then statement 3 must be false, making statement 1 true, and thus statement 2 would imply its falsity and its falsity would imply its truth. Since this is not possible, it is a logical paradox, showing that they are not statements after all

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