Analysis problem: x>o -> 1/x > 0

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In summary, the conversation discusses the analysis problem of proving that if x>0, then 1/x>0 using the ordered field axioms. The attempt at a solution involves using a direct proof and the existence of inverse, but ultimately leads to a dead end. The other user suggests using the property of things less than 0, and the conversation concludes with a proof that involves assuming 1/x is less than or equal to 0 and leads to a contradiction, ultimately proving that 1/x>0.
  • #1
b0it0i
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Analysis problem: x>o --> 1/x > 0

Homework Statement


Prove

If x>0 --> 1/x > 0


Homework Equations



ordered field axioms

closure, associativity, commutativity, identity, inverses, distributive law, trichotomy law, transitive law, preservation

x+z = y+z --> x = y
x.0 = 0
-1.x = -x
xy=0 iff x=0 or y=0
x<y iff -y<-x
x<y and z<0 then xz > yz


The Attempt at a Solution



i've tried this problem several times, and always hit a dead end

i tried a direct proof

assume x>0
therefore x does not equal 0

by existence of inverse

there exists a unique 1/x such that x (1/x) = 1

after that point, i get no where in my attempts

any suggestions?

you can user other "theorems" but you must also prove them
 
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  • #2
Assume 1/x<0. Can you derive a contradiction using one of the properties you listed? (Hint: which one deals with things <0?)
 
  • #3
hey thanks a lot

prove: x > 0 --> 1/x > 0

assume 0 < x and assume 1/x less than or equal to 0, x cannot equal 0. then there exists a unique 1/x s.t. x(1/x) = 1. since 1/x is unique, 1/x cannot equal zero.
therefore 1/x < 0

since 0 < x and 1/x < 0

0. 1/x > x . 1/x
0 > 1

which contradicts 0 < 1

i would have to prove that 1 > 0, but i already done this.

thanks
 

Related to Analysis problem: x>o -> 1/x > 0

1. What is the analysis problem in the statement "x>0 -> 1/x > 0"?

The analysis problem in this statement is to determine what conditions must be met for the statement to be true.

2. How can I analyze the given statement to determine if it is true?

In order to analyze the statement, you can use mathematical principles and logical reasoning to determine if the conditions are met for the statement to be true.

3. Can you explain the significance of the symbols used in the statement?

The symbol ">" represents "greater than", "x" represents a variable, "0" represents zero, and "->" represents "implies". These symbols are commonly used in mathematical statements to represent relationships and logical implications.

4. Are there any exceptions to the statement "x>0 -> 1/x > 0"?

Yes, there are exceptions to this statement. For example, if x is equal to 0, then the statement is not true because dividing by 0 is undefined.

5. How is this analysis problem relevant in the field of science?

This analysis problem is relevant in science because it involves using logic and reasoning to analyze a statement and determine its validity. This type of critical thinking is essential in scientific research and experimentation.

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