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 n 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.