Infected Integers...
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2 solutions
The inequality does not make any sense for z < 0 .
If I were to ask the question as which of these is valid for integers z < − 1 with options as
A. z ≤ z 2
B. z ≤ z 2
C. z = z 2
Which is the correct option?
Can anyone help me on that?
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none of the options
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If it is'none of this', then it would mean that the relational operator has no meaning while dealing with complex numbers. 2 > 1 > − 1 . But, it is meaningless to say − i < i or i < − i .
The relationship z ≤ z 2 is 'meaningless for z < 0 ', not invalid.
3 z < 3 z 3 is invalid for z ∈ ( − ∞ , − 1 ) ∪ ( 0 , 1 ) because the inequality still makes sense in the said domain. But, the previous one does not.
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@Janardhanan Sivaramakrishnan – The complex numbers cannot be ordered. See this note for a reason.
It is not guaranteed that we can always compare 2 elements. This is only possible if the structure we are working with is totally ordered .
@Janardhanan Sivaramakrishnan – I couldn't follow the point you wanted to make.
Okay, I went through your comment again.
So are we dealing with the difference between the words "invalid" and "meaningless"?
Moreover, the question states the full domain of Integers (positive as well as negative). So the inequality is meaningful for the positive numbers and meaningless (not meaningful) for the negative numbers. Why can't I say that it's valid for positive numbers and invalid (not valid) for negative numbers.
I am not so thoroughly acquainted with terminology used in higher maths. But if we cannot say that "invalid" and "meaningless" are same things as per the context of our problem. Then I don't mind asking a moderator to change the word 'valid' with 'meaningful' in the options.
:)
Not exactly. It holds true for all integers greater than or equal to 0 .
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No where in the question did we mention that we are only dealing with integers greater than or equal to 0. Niranjan Khanderia is correct.
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I know he is right. I was pointing out that his solution failed to include z=0 earlier. He added that now though .
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@Shashank Kancherla – oops..sorry I overlooked it.. :)
Thanks. Technically I was correct. But it would be better if 0 is included. It seems that I had put 1 instead of 0 absentmindedly.
i thought the 2 options said same things, i am wrong though.
Do you mean "not valid for z < 0 "?
in the case of (0, 1) and (1, infinity)!
It is true for z ≥ 0, and not valid for z < 0.