Is there a simple injection from B to A using tangent function?

In summary: B...to C...to D...to E...to F...to G...to H...to I...to J...to K...to L...to M...In summary, In order to find a function that goes from A to B, we need to map points (0,0), (2pi,0), (4pi,0), (6pi,0), (8pi,0), (10pi,0), (12pi,0), (14pi,0), (16pi,0), (18pi,0), (20pi,0), (22pi,0), (24pi,0), (26pi,0), (28pi,0), (30
  • #1
Homework Statement
Find a bijection between A and B.
Relevant Equations
A = (−∞, 0) ∪ (0, ∞) ⊂ R,
B = {(x, y) ∈ R2| x2 − y2 = 1}
So I don't really have any goods ideas on how to try and solve this. I only know that using tangent function is supposedly a good idea.
 
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  • #2
I think that if you isolate ##y##, you can make a piecewise injective function of ##x## that reaches all values of ##A##.
 
  • #3
Tjaz Gantar said:
Homework Statement: Find a bijection between A and B.
Homework Equations: A = (−∞, 0) ∪ (0, ∞) ⊂ R,
B = {(x, y) ∈ R2| x2 − y2 = 1}

So I don't really have any goods ideas on how to try and solve this. I only know that using tangent function is supposedly a good idea.

You have ##\sec^2 t - \tan^2 t = 1## and ##\cosh^2 t - \sinh^2 t = 1##.

Either of these might be a good starting point.

Have you drawn a graph of B?
 
  • #4
PeroK said:
You have ##\sec^2 t - \tan^2 t = 1## and ##\cosh^2 t - \sinh^2 t = 1##.

Either of these might be a good starting point.

Have you drawn a graph of B?
Can't we consider ##f(x) := \sqrt{x^2-1}## for ##x>1## and ##-\sqrt{x^2-1}## for ##x<-1##?
 
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  • #5
archaic said:
Can't we consider ##f(x) := \sqrt{x^2-1}## for ##x>1## and ##-\sqrt{x^2-1}## for ##x<-1##?

Okay, where are you mapping the following 4 points in B?

##x = 2, y = \sqrt{3}; \ x = 2, y = -\sqrt{3}; \ x = -2, y = \sqrt{3}; \ x = -2, y = -\sqrt{3}##
 
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  • #6
archaic said:
Doesn't a bijection mean that the function is both injective and surjective?

Yes. I note that this is not your homework!

B is not the graph of a function.
 
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  • #7
Tjaz Gantar said:
Homework Statement: Find a bijection between A and B.
Homework Equations: A = (−∞, 0) ∪ (0, ∞) ⊂ R,
B = {(x, y) ∈ R2| x2 − y2 = 1}

So I don't really have any goods ideas on how to try and solve this. I only know that using tangent function is supposedly a good idea.
Hello @Tjaz Gantar .
:welcome:

I see that although you have been a member for more than half a year, this is your first post here at PF.

Please show more specifics regarding your understanding of this problem, and what you have considered in its analysis.

As @PeroK suggests, draw a graph of B.

By the way; do you mean to say x and y are squared in the definition of set B as in the following?
B = {(x, y) ∈ ℝ2| x2 − y2 = 1}​
.
 
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  • #8
PeroK said:
Okay, where are you mapping the following 4 points in B?

##x = 2, y = \sqrt{3}; \ x = 2, y = -\sqrt{3}; \ x = -2, y = \sqrt{3}; \ x = -2, y = -\sqrt{3}##
Hey, I hope I'm not intruding on the HW further.
Isn't it irrelevant whether we map every element of the starting set if every element of the final has an antecedent? The function I gave takes all ##x##s in ##B## (##x\in\mathbb{R}-[-1,1]##) and gives you ##A##.
 
  • #9
archaic said:
Hey, I hope I'm not intruding on the HW further.
Isn't it irrelevant whether we map every element of the starting set if every element of the final has an antecedent? The function I gave takes all ##x##s in ##B## (##x\in\mathbb{R}-[-1,1]##) and gives you ##A##.

I don't know what that means.
 
  • #10
archaic said:
Hey, I hope I'm not intruding on the HW further.
Isn't it irrelevant whether we map every element of the starting set if every element of the final has an antecedent? The function I gave takes all ##x##s in ##B## (##x\in\mathbb{R}-[-1,1]##) and gives you ##A##.
Yes, it's relevant. You need the map to be both one to one and onto. a bijection.
 
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  • #11
PeroK said:
I don't know what that means.
SammyS said:
Yes, it's relevant. You need the map to be both one to one and onto. a bijection.
I thought I remembered the definition correctly, my bad!
 
  • #12
archaic said:
I thought I remembered the definition correctly, my bad!

Take an example where ##B## is defined by ##x^2 + y^2 = 1##. Then we can define an injection from ##[0, 2\pi)## to ##B## using ##x = \cos t, \ y = \sin t##.

If ##B## were one half of a hyperbola, then the problem would be equally simple. But, ##B## is the two disjoint halves of the hyperbola. That makes things a little trickier.

But note that ##A## also is disjoint.

Another idea that may be useful is that the composition of two injections is a injection.
 
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  • #13
PeroK said:
Take an example where ##B## is defined by ##x^2 + y^2 = 1##. Then we can define an injection from ##[0, 2\pi)## to ##B## using ##x = \cos t, \ y = \sin t##.

If ##B## were one half of a hyperbola, then the problem would be equally simple. But, ##B## is the two disjoint halves of the hyperbola. That makes things a little trickier.

But note that ##A## also is disjoint.

Another idea that may be useful is that the composition of two injections is a injection.
Are we trying to go from ##A## to ##B##?
 
  • #14
archaic said:
Are we trying to go from ##A## to ##B##?

A bijection, by definition, goes both ways.
 
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  • #15
archaic said:
Are we trying to go from ##A## to ##B##?
Actually, there is an injection from ##B## to ##A## which is quite remarkable. I'm not sure what the simplest way to find it is. It's the inverse of the one I found going from ##A## to ##B## starting with the hyperbolic trig parameterisation of ##B##.

It's definition is absurdly simple and looks unbelievable.
 
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  • #16
PeroK said:
An injection, by definition, goes both ways.
Presumably you meant to say: a bijection, by definition, goes both ways.
 
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  • #17
As @PeroK pointed out earlier, if we look at the graph of the second set, we find that it is a hyperbola. So, we see hyperbola has two branches, and our set A, also has two 'branches', that is, two disjunct intervals. Also, both of these are in a sense symmetrical. Hyperbola is symmetrical with respect to ##y##-axis, while set A is symmetrical with respect to 0. So the idea could be to create parametrization of each branch of the hyperbola onto one of those intervals. By symmetry, the other branch could be parametrised trivially after this has been done. So you're searching for a function that maps positive real numbers into the positive(##x>0##) branch of the hyperbola, for example.
 
  • #18
SammyS said:
Hello @Tjaz Gantar .
:welcome:

I see that although you have been a member for more than half a year, this is your first post here at PF.

Please show more specifics regarding your understanding of this problem, and what you have considered in its analysis.

As @PeroK suggests, draw a graph of B.

By the way; do you mean to say x and y are squared in the definition of set B as in the following?
B = {(x, y) ∈ ℝ2| x2 − y2 = 1}​
.
Well I understand bijection and know what i need to do, I just had no idea how to make it so that it really is a bijection if you understand what i mean by that:D
And yes, its ment to be squared, didn't notice that it was written like that
 
  • #19
Tjaz Gantar said:
Well I understand bijection and know what i need to do, I just had no idea how to make it so that it really is a bijection if you understand what i mean by that:D
And yes, its ment to be squared, didn't notice that it was written like that

I don't know if there is an easier way to get a solution, but what I did was:

Start with the parameterisation of ##B## using the hyperbolic trig functions. Then, remember that the composition of two bijections is a bijection.
 
  • #20
Antarres said:
As @PeroK pointed out earlier, if we look at the graph of the second set, we find that it is a hyperbola. So, we see hyperbola has two branches, and our set A, also has two 'branches', that is, two disjunct intervals. Also, both of these are in a sense symmetrical. Hyperbola is symmetrical with respect to ##y##-axis, while set A is symmetrical with respect to 0. So the idea could be to create parametrization of each branch of the hyperbola onto one of those intervals. By symmetry, the other branch could be parametrised trivially after this has been done. So you're searching for a function that maps positive real numbers into the positive(##x>0##) branch of the hyperbola, for example.
So if i make it
x=t (t being element of A), y=|√(x2 - 1)| for t≥1
and
y=tan(π(t-1)/2), x=|√(y2+1)| for 0<t<1
would that be okay from "the right part" of A to "the right part" of the hyperbola?
 
  • #21
Tjaz Gantar said:
So if i make it
x=t (t being element of A), y=|√(x2 - 1)| for t≥1
and
y=tan(π(t-1)/2), x=|√(y2+1)| for 0<t<1
would that be okay from "the right part" of A to "the right part" of the hyperbola?

Yes, that looks right.

Note that ##\sqrt{x}## is non-negative, by definition.
 
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  • #22
PeroK said:
Yes, that looks right.
okay its nice to see progress in my head hah thanks
 
  • #23
In theory we can also use Schroeder Bernstein and produce injections from each set to the other. Not likely but it may have been used in the class without proof.
 
  • #24
PeroK said:
Actually, there is an injection from ##B## to ##A## which is quite remarkable. I'm not sure what the simplest way to find it is. It's the inverse of the one I found going from ##A## to ##B## starting with the hyperbolic trig parameterisation of ##B##.

It's definition is absurdly simple and looks unbelievable.
Is this that injection from ##B## to ##A##
##f(x,y)=x+y##
?
 
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1. What is a tangent function?

A tangent function is a mathematical function that describes the relationship between the sides of a right triangle. In other words, it helps us find the ratio of the opposite side to the adjacent side in a right triangle.

2. How is the tangent function used in injections from B to A?

The tangent function can be used to map values from set B to set A in an injective (one-to-one) manner. This means that every element in set B is uniquely mapped to an element in set A, and no element in set B is left unmapped.

3. Is there a simple injection from B to A using only the tangent function?

It is not possible to create a simple injection using only the tangent function. Other mathematical operations and functions would also be needed to create an injection from set B to set A.

4. Can the tangent function be used to create a surjective (onto) mapping from B to A?

No, the tangent function cannot be used to create a surjective mapping from set B to set A. This is because the tangent function has a limited range of values and cannot cover all elements in set A.

5. Are there any other mathematical functions that can be used to create an injection from B to A?

Yes, there are many other mathematical functions that can be used to create an injection from set B to set A. Some examples include logarithmic functions, exponential functions, and polynomial functions.

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