Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

- Index
- » Search
- »
**Posts by Kazy**

**Kazy**- Replies: 1

I need to prove the following:

a) Let A, B be sets contained in a universal set U. Suppose that A ≈B. prove that P(A) ≈ P(B) (power sets)

b) Let A, B be sets contained in a universal set U such that A is a subset of B. Suppose that A is countable and B is uncountable. prove that B - A is uncountable.

c) Using the Schroeder-Bernstein Theorem, prove that any two intervals of real numbers are numerically equivalent.

Shroeder-Bernstein Theorem: Let A and B be sets, and suppose that A <= B and B <= A. Then A ≈ B.

Can anyone help with any of those?

**Kazy**- Replies: 1

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?

Let A and B be sets contained in some universal set U. We say that A and B are numerically equivalent if there exists a bijection f: A->B.

That help?

**Kazy**- Replies: 3

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?

**Kazy**- Replies: 1

I need to prove:

a) Prove that if S is any finite set of real numbers, then the Z U S (integers Union S) is countably infinite.

b) Let A, B, C, D be subsets of a universal set U. Suppose that A ≈ B and C ≈ D. Prove that A X C ≈ B X D (cartesian products)

c) Let A and B be sets contained in a universal set U. Suppose there exists an injection f: A -> B. Prove that if A is countably infinite, then B is countable.

For A, I'm thinking Z U *anything* is countably infinite since Z is infinite. I'm not sure if thats all I have to say, or even really how to make that a formal proof. For B and C I don't even know where to begin.

Can anyone help me out? Thanks!

**Kazy**- Replies: 1

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?

**Kazy**- Replies: 1

I need to the prove the following:

Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|.

This is confusing me because doesn't the definition of numerically equivalent actually say that the cardinality of A = the cardinality of B? If so, does that mean I can just say "By the definition of numerically equivalent"?

Thanks!

**Kazy**- Replies: 2

I need to prove that the cubed root of 2 is irrational. How would I go about that?

Also give this statement: "If n is any positive integer, then n² + n + 41 is always a prime number." I need to either prove it correct or give a counterexample. From looking at it I thought there was no way that was true, however every number I plug in works out so i'm guessing its probably true. However i have no idea how to prove that it is true. Any help?

**Kazy**- Replies: 1

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.

**Kazy**- Replies: 1

I need to prove:

a) Let a, b, c ∈ Z. Suppose that a | c and b | c and that gcd(a,b) = 1. Prove that ab | c.

b) prove that for every integer n, gcd(n, n+1) = 1

Both of these seem trivial and are obviously true, but other than saying "These are obviously true" I don't even know where to begin. It seems I have more trouble with proofs when I know their easy and true, then when i'm not sure if their true. Can anyone help?

**Kazy**- Replies: 5

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?

**Kazy**- Replies: 10

I need to prove the following:

a) Let a, b, c, d ∈ Z. Prove that if a < b and c < d then a+c<b+d.

b) Suppose that ab > 0. Prove that either a > 0 and b > 0 or a < 0 and b < 0.

These are both obviously true, and yet its so simple to just look at I can't think of how to "prove" it.. Any help?

**Kazy**- Replies: 1

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!

**Kazy**- Replies: 1

I need to prove that multiplication on Q is well defined. I have no idea really what well defined even means. From what I understand from reading, it means that multiplication on Q does not depend on the choices of integers to represent the equivalence classes. However, I have no idea how to go about proving that. Anyone have any ideas?

Thanks!

**Kazy**- Replies: 3

I need to prove the following:

a) Addition on Q is communitive.

b) Multiplication on Q is associative.

c) Multiplication on Q is communitive.

I figured out how to prove addition on Q is associative.. Here is what I did:

([a,b] + [c,d]) + [e,f]

= [ad + bc, bd] + [e,f] = [(ad+bc)f + (bd)e, (bd)f]

= [adf+bcf+bde, bdf] = [a(df) + b(cf+de), b(df)]

= [a,b] + [cf + de, df] = [a,b] + ([c,d] + [e,f])

I can't figure the others out. any help would be appreciated.

**Kazy**- Replies: 1

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.

**Kazy**- Replies: 1

I'm supposed to use the Lagrange interpolation process to obtain a polynomial of least degree that assumes these values:

X| 0 | 2 | 3 | 4

Y| 7 | 11 | 28 | 63

My book just tells the form with a bunch of symbols and gives no examples.. I'm completely lost. Any help? Thanks!

**Kazy**- Replies: 4

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.

**Kazy**- Replies: 2

I need to prove the following:

1) Let f ∈ F(A,B). Prove that if f is bijective, so is f-¹

2) Let f ∈ F(A,B) and g ∈ F(B,C). prove that if f and g are invertible so is gf and that (gf)-¹ = f-¹g-¹

3) Let A,B, and C be nonempty sets, and let f ∈ F(A,B) and g, h ∈ F(B,C)

a) prove that if f is surjective and gf=hf, then g=h.

b) Give an example in which gf=hf, but g ≠ h

I'm completely lost on those 3. Any help would be appreciated.

Thanks! I appreciate it.

or did you mean f-¹ as the inverse image?

- Index
- » Search
- »
**Posts by Kazy**