Simplify Set Notation Operations?

Question

Hi, I'm Sara.

I had this exercise that I'm not too sure about:

simplify {4, 7, 8, 9} ∪ ({1, 2, 3, ...} ∩ {2, 4, 6, ...}

Here are my attempts.

I simplified what is inside the parenthesis and got {2, 4, 6, ...} then I got stuck on this:

{4, 7, 8, 9} ∪ {2, 4, 6, ...}

I got 3 possible answers, but I'm still not sure about them:

1. {2, 4, 6, 7, 8, 9, ...}

2. {2, 4, 6, 7, 8, 9, 10, 12, 14, ...}

3. {positive even integers and 7 and 9}

I'm leaning towards the 3rd one, but I am still not sure. Please help.

Much appreciated.

Answer 1

You are correct on the third one.

First it's the intersection, so every matching number stays, which are only the even numbers.  Then it's union, which combines the two, which are all even numbers and 7 and 9.

I also think however that the second one might be correct notation, but I can't say whether you can put phrases inside or not.

Answer 2

You need to perform the operations inside out using the parentheses.

{4, 7, 8, 9} ∪ ({1, 2, 3, ...} ∩ {2, 4, 6, ...})

Start with

{1, 2, 3, ...} ∩ {2, 4, 6, ...}

this is equivalent to

{2, 4, 6, ...}

so....

{4, 7, 8, 9} ∪ {2, 4, 6, ...}

Simply combine the sets (union means combine)

{ 7, 9, positive even integers}

Number 3 is the best way to answer the question.