Operation on Sets Homework: B-A & C-A

[ver_mathstats]

Homework Statement

Let A= {1, 2, 3}, B= ℤ⁺, C= [1, infinity)

That is C= {x∈ℝ:x≥1}

What is B - A and C - A?

Homework Equations

The Attempt at a Solution

I am unsure of how to go about answering this.

I know that B - A means what elements are in B that aren't in A. Would that make the answer B - A = {Z⁺>3}? Or is B - A an empty set since 1, 2, 3 are positive integers?

As for C - A, I am unsure of how to approach this one, would it be C - A = {x > 3}?

Would both just be greater than 3 since A = {1, 2, 3}?

 

[fresh_42]

Homework Statement

Let A= {1, 2, 3}, B= ℤ⁺, C= [1, infinity)

That is C= {x∈ℝ:x≥1}

What is B - A and C - A?

Homework Equations

The Attempt at a Solution

I am unsure of how to go about answering this.

I know that B - A means what elements are in B that aren't in A. Would that make the answer B - A = {Z⁺>3}?

Correct, although I would either write it $\mathbb{N}_{>3}$ or $\{\,4,5,6,\ldots\,\}$

Or is B - A an empty set since 1, 2, 3 are positive integers?

No. B-A are all natural numbers (B) which are not 1,2 or 3 (A). How do you get an empty set?

As for C - A, I am unsure of how to approach this one, would it be C - A = {x > 3}?

No, since e.g. 5/2 is still in C and therewith in C-A. Draw C, take scissors and cut out 1,2,3.

Would both just be greater than 3 since A = {1, 2, 3}?

No, you only take away the three points, not those in between.

 

[ver_mathstats]

Correct, although I would either write it $\mathbb{N}_{>3}$ or $\{\,4,5,6,\ldots\,\}$
No. B-A are all natural numbers (B) which are not 1,2 or 3 (A). How do you get an empty set?
No, since e.g. 5/2 is still in C and therewith in C-A. Draw C, take scissors and cut out 1,2,3.

No, you only take away the three points, not those in between.

I got confused about the empty set but I know now it is natural numbers greater than 3. As for C - A I understand what you are saying now, however I am struggling to write it out. C - A = {x ε ℝ⁺ : x ≠ 1, 2, 3}, would this be okay?

Thank you.

 

[RPinPA]

I got confused about the empty set but I know now it is natural numbers greater than 3. As for C - A I understand what you are saying now, however I am struggling to write it out. C - A = {x ε ℝ⁺ : x ≠ 1, 2, 3}, would this be okay?

Thank you.

Not quite, since C doesn't include any numbers < 1 and your set does. Restrict those and you're fine.
I've got this nagging feeling that there's something you're still missing about the definition of these sets and of the subtraction operation.

 

[ver_mathstats]

Not quite, since C doesn't include any numbers < 1 and your set does. Restrict those and you're fine.
I've got this nagging feeling that there's something you're still missing about the definition of these sets and of the subtraction operation.

Oh I thought that if I put the ℝ then ⁺ that would make it clear it was positive only but I will try to write it out differently. I am new to this and I think just the fact C is [1, infinity) through me off a little. But thank you for the help.

 

[RPinPA]

Oh I thought that if I put the ℝ then ⁺ that would make it clear it was positive only but I will try to write it out differently. I am new to this and I think just the fact C is [1, infinity) through me off a little. But thank you for the help.

It does mean that it is positive. For instance 0.5 is a positive number which is included in $R^+$. It's a real number, and it's positive. But it's not in C, which does not have any numbers less than 1.

 

[ver_mathstats]

It does mean that it is positive. For instance 0.5 is a positive number which is included in $R^+$. It's a real number, and it's positive. But it's not in C, which does not have any numbers less than 1.

Oh right, I am very sorry for missing that, I get what's wrong now. Thank you so much.

 

[fresh_42]

I am struggling to write it out

You cannot simplify much here, so something like $\neq 2,3$ cannot be avoided. What you can do, is to modify the interval here:
What is $C - \{\,1\,\}\,?$.
Then you can write $\left( C- \{\,1\,\} \right) - \{\,2,3\,\}$ or $\{\,x \in \mathbb{R}\,|\,\ldots \,\}$.