Theory of Inequalities (Whole Integers)

[petert]

Currently studying inequalities and transformations, when I came up with this theory for real whole integers.

For all real whole numbers a, b, and c, when a=a, b=b, and c=b-1, the following is true:

if a<b then a<c

same goes here:

if a>c then a>b

Can you find a number for me that disproves this?

Edit: This sign (<) means less than or equal to. I couldn't find the proper symbol. The opposite goes for this (>), which means greater than or equal to.

[JGR]

Currently studying inequalities and transformations, when I came up with this theory for real whole integers.

For all real whole numbers a, b, and c, when a=a, b=b, and c=b-1, the following is true:

if a<b then a<c

same goes here:

if a>c then a>b

Can you find a number for me that disproves this?

You don't really need "c" at all.
Also the relationship works both ways (AFAICT), so you can say:

if and only if a<b then a<b-1
if and only if a>b-1 then a>b

Which is fairly obvious for integers, and indeed there are no integers which would disprove it.

[CommanderZ]

That is kinda stupid in my eyes.

a doesn't participate in the relationship c=b-1, so we (according to definition of the set of integers) can always find a value for which both inequalities are true.

EDIT: This gotta work for reals exactly as well. The first set where it would fail could be R*.

[petert]

a doesn't participate in the relationship c=b-1, so we (according to definition of the set of integers) can always find a value for which both inequalities are true.

EDIT: This gotta work for reals exactly as well. The first set where it would fail could be R*.

A is just a variable, which is necessary here because of the inequality. And yes, if c=b-1, then a doesn't participate, but when comparing a to b and c, it is required. Does this make sense? Even I'm getting a bit confused.

Also, what is R*?

[JGR]

IIRC, integers is a subset of real, and he did say integers (and indeed real) in his post, so he didn't claim it would work for complex (or indeed non-whole reals), which it would not.

A lot of maths is technically true, but fairly useless (I suppose this exercise could help in learning the basics of inequalities operators, maybe?)


Frankly there's no point in even trying to look at those inequalities outside the integer set, they fail quite obviously.

[CommanderZ]

Also, what is R*?

Extended reals set, real + positive infinity + negative infinity.

so he didn't claim it would work for complex (or indeed non-whole reals), which it would not.

It would fail for complex numbers, since no such thing like ">" and "<" applies for them.

[petert]

Thanks for all the help. Now I guess we can change this from the theory of inequalities to the property of inequalities.