Equivalence Problem: Is A and B Equivalent?

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Homework Help Overview

The discussion revolves around the equivalence of two statements regarding integers and their square roots. Statement A asserts that an integer n is not a perfect square, while Statement B claims that if n is greater than or equal to 1, then the square root of n is either an integer or irrational.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express differing views on the equivalence of the two statements, with some arguing that they are not equivalent due to the nature of the assumptions and implications of each statement. Questions arise about the definitions and interpretations of the statements, particularly regarding the conditions under which the square root is considered.

Discussion Status

The discussion is ongoing, with participants actively exploring the distinctions between the two statements. Some have provided reasoning to support their views on non-equivalence, while others have acknowledged the need for clarity in understanding the implications of each statement.

Contextual Notes

Participants note that Statement B includes the condition n >= 1, which influences the interpretation of the square root's nature. There is an emphasis on the need to consider specific examples to illustrate the arguments being made.

soopo
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Homework Statement


Are the statements A and B equivalent with each other?

A. Suppose that n is an integer which is not a perfect square.
B. If n >= 1, then \sqrt{n} is either an integer or is irrational.

The Attempt at a Solution


I am keen on saying that the two statements are not equivalent.
However, Oxford's undergraduate booklet claims that they are equivalent.

In my opinion, A is inclined to that n is not a perfect square, while B is neutral.
 
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they are quite different. Statement 1, is an assumption (which we are assuming to be true), whereas statement 2 is a...well its a statement (that may be true/false)

Also, consider the second statement

It says sqrt(n) is either integer or irrational. This means sqrt(n) could be sqrt(2/3) which is also irrational. However, it is not equivalent to saying n is an "integer" and not a perfect square..

edit: ok, Statement 2 also says n>=1(missed that) so just do the entire same argument for sqrt(3/2), that'll also hold
 
they are quite different. Statement 1, is an assumption (which we are assuming to be true), whereas statement 2 is a...well its a statement (that may be true/false)

Also, consider the second statement

It says sqrt(n) is either integer or irrational. This means sqrt(n) could be sqrt(2/3) which is also irrational. However, it is not equivalent to saying n is an "integer" and not a perfect square..
 
praharmitra said:
they are quite different. Statement 1, is an assumption (which we are assuming to be true), whereas statement 2 is a...well its a statement (that may be true/false)

Also, consider the second statement

It says sqrt(n) is either integer or irrational. This means sqrt(n) could be sqrt(2/3) which is also irrational. However, it is not equivalent to saying n is an "integer" and not a perfect square..

I agree with you.
 
praharmitra said:
edit: ok, Statement 2 also says n>=1(missed that) so just do the entire same argument for sqrt(3/2), that'll also hold

Do you mean that the two statements are equivalent?
 
No he still means they are not equivalent. Just plug in 3/2 instead of 2/3 in his previous argument of why they are not equivalent.
 
dperkin2 said:
No he still means they are not equivalent. Just plug in 3/2 instead of 2/3 in his previous argument of why they are not equivalent.

Thank you both!
The problem is now clear.
 

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