Math Is Fun Forum

I need to prove that the following sets are countably infinite:

a) Q (intersect) [0,1]   -  Rationals intersected with [0,1]
b) Q+ U {e^x | x ∈ Z}
c) The rational points on the unit circle:
    {(x,y) | x² + y² = 1, x ∈ Q, y ∈ Q}

I know c has to do with pythagorean triples, but other than that, i'm lost. Can anyone help?

I need to prove the following:

a) Prove that every closed interval [a,b] is numerically equivalent to [0,1].
b) Prove that any two closed intervals [a,b] and [c,d] are numerically equivalent.
c) Prove that any two open intervals (a,b) and (c,d) are numerically equivalent.
d) Let [a,b] be a closed interval and (c,d) be an open interval. Prove that [a,b] and (c,d) are numerically equivalent.

I'm guessing all of these are proven pretty much the same way, but I don't even know where to start. Can anyone help?

I need to prove the following:

a) Prove that the function f(n)={  2n, if n>0  and -2n+1, if n ≤ 0} is bijective.
b) Prove that Z ≈ Z+ by finding a bijective function g: Z+ -> Z.
c) Let Z- be the set of negative integers. Prove that Z- ≈ Z+ by finding a bijective function f: Z+ -> Z=. prove that your function is bijective.

I know to prove something is bijective you need to show its surjective and injective, however I'm completely lost from there. Any help?

I need to prove the following about prime numbers:

a) Let n ∈ Z, n > 1. Prove that if n is not divisible by any prime number less than or equal to √n, then n is a prime number.
b) Let n be a positive integer greater than 1 with the property that whenever n divides a product ab where a, b ∈ Z, then n divides a or n divides b. Prove that n is a prime number.
c) Prove that 2 is the only prime of the form n^3 + 1.

Can anyone help with any of those? Thanks.

Given this theorem:
Let a,b ∈ Z, not both equal. Then a greatest common divisor d of a and b exists and is unique. Moreover, there exist integers x and y such that d = ax + by. 

I need to prove the unique part of the theorem. I figured out the proof of the existance, but I have no idea how ot prove that the gcd of a and b is unique.

After that, I also need to prove the following:
Let a, b ∈ Z, not both zero. Let S = {n∈Z | n=ax + by, for some x, y ∈ Z}. Let d = (a,b) where (a,b) is the g.c.d of a and b. Prove that S is the set of all integer multiples of d.

I used the set S in the previous part to prove the first part of the theorem, and I'm guessing i need to use the theorem and manipulate it somehow to prove that S is the set of all integer multiples of d.

Can anyone help?

I need to prove that every element of Q (rational numbers) has an inverse with respect to addition. From what understand, I need to use the fact that [0,1] is the additive identity of Q. However, I have no idea what to do beyond that.

Also, I need to prove that every element of Q except [0,1] has an inverse with respect to multiplication. Again, to do so, I'm thinking it has something to do with the fact that [1,1] is the miltiplicative identity of Q.

If anyone can help, that'd be great!

I need to prove that R is an equivalence relation S in the following:

S = {(a,b) ∈ Z x Z | b ≠ 0}. R is the relatoin on S defined by (a,b)R(c,d) if ad=bc.

Can anyone help? I'm completely lost.

I need to determine which of the following have identities with proofs.
a) On Z (integers), n * m = 6 + nm
b) On Z, n * m = n²m²
c) On Z+, n *m = min(n,m), the smaller of n and m
d) On P(A), for any set A, X * Y = X U Y
e) On Z+, n * m =  n^m

I understand the concept of identities, i'm just unsure on how you can prove these things. I know if they don't have an identity, its easiest to show it by using a contradiction, but I'm having trouble finding any contradictions. Any help would be appreciated! Thanks.