How do i write these sets of real numbers write the EUF and EnF ?

Question

If E={yly<=3}and F= {yly>6} how do I write the union and the intersection using interval notation?

Answer 1

if the "=" sign is included use the bracket [ or ]

if the "=" sign is not included use the parenthesis ( or )

E contains ALL numbers less than 3 including 3 (see the = sign ?)

F contains ALL numbers greater than 6 but not 6 itself (there is no = sign)

you can't write all the numbers greater than 6 so you use the infinity symbol (∞ or -∞) as the "last" number in the interval but you cannot ever get to ∞ or -∞ so that's why you use ( or ) instead of [ or ]

let's assume that you are working with only integers for the sake of understanding, but remember you are talking about

ALL numbers as indicated by the intervals

E=(-∞,...,-4, -3, -2, -1, 0, 1, 2, 3]

F=(6, 7, 8, 9, 10, 11, ...∞)

E and F  have no elements in common so the union is all the elements in E plus all the elements in F:

EUF=(-∞,3]U(6,∞)

because the two sets have no elements in common the intersection is the null set or empty set denoted by { } or Ø

again, don't forget you are using ALL numbers , not just integers

look at this example:

A={yly<7} and B={yly≥2}

again assuming only integers...

A=(-∞,...-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7)

B=[2, 3, 4, 5, 6, 7, 8, ...∞)

what's missing from A ? 7, 8, 9, 10, 11, ...up to ∞, but these are all in B

therefore AUB=(-∞,...-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...∞)=(-∞,∞)

what do A and B have in common ? 2, 3, 4, 5, 6 but not 7

A∩B=[2, 3, 4, 5, 6]=[2,6]

Answer 2

Before going through the "answer" I just want to be sure you're reading the set notation properly - otherwise that will surely throw you off too.

 

E={y|y<=3} essentially is read, "The set of all y such that y is less than or equal to 3." The vertical line means "such that" is where I get that from.

 

My previous answer had the descriptions of union and intersection, but in case that wasn't clear I'll certainly try again.

 

EυF is the union, or joining of the two sets. A union in essence puts the sets together and combines them. If I have a couple of sets of something a lot more manageable than real numbers, like words, it might look like:

 

E={cat, dog, fish} and F={fish, monkey, dog}.

 

The union of sets puts them together, but note, it's without duplicates. So now,

EυF = {cat, dog, fish, monkey}.

 

The intersection of two sets, again like with roads, describes what they share in common. Where two roads intersect, they have common pavement. In my word example the intersection of the sets is the words that appear in both sets. so E∩F = {dog, fish}.

 

So now with numbers, and the question you have is just written in interval notation, but it has an annoyance your first problem didn't when trying to figure out the intersection. Set E is, as I said above, all the numbers less than or equal to 3, while set F is all numbers greater than 6. Notice that the two sets don't share any common numbers, so the intersection is the empty set.

 

The union of the two sets has a gap. It's all the numbers from 3 down, and all the numbers bigger than 6. So, in interval notation:

 

(-∞, 3] υ (6,∞)

Answer 3

In interval notation, E=(-∞,3], and F=(6,∞).

 

For the union, we just put both intervals on the number line together. Since these two don't touch or overlap, the simplest way to write that is just: (-∞,3] ∪ (6,∞).

 

The intersection is the overlap of the two. These two intervals have no overlap, so their intersection is the empty set, which you might denote as {}, or as ∅.