What is the answer of X' cap Y

What is the answer of Y' U Z'
What is the answer of (X \ Z ) '
What is the answer of ( Y \ Z ) ' U X '
X={10 ,20 ,30 ,40 ,50}
Y={30 ,50, 60, 90, 100}
Z={40, 50, 60, 70, 80}
U={200, 300}

When you deal with complements, the first question to ask is

do we know the universal set E?

Here we assume E=Z (integers)

X'∩Y = Y\X = {10,20,40,60,90,100}
Y'&cup'Z'= YΔZ= {30,40,70,80,90,100}
(X\Z)'={10,20,30}' depending what E equals
(Y\Z)'∩X'
=((Y\Z)∪X)'
=({30,90,100}∪{10,20,30,40,50})'
={10,20,30,40,50,90,100}'
depending on what E is.

(note U cannot be the universal set because it does not contain any of the other members).

Note:
you can type
∩ as & c a p ;
∪ as & c u p ;
(do not enter interstitial spaces)

*

Y'∪Z'= YΔZ= {30,40,70,80,90,100}

Thank you

To find the answer for each of the given expressions, we will use the set operations and apply them to the given sets X, Y, and Z.

1. X' cap Y:
To find X' (complement of X), exclude the elements of X from the universal set U. In this case, the universal set U does not seem to be defined, so let's assume that U is the set containing all the numbers up to 1000. So, X' = U \ X, which means X' includes all the numbers not present in X.

X' = {10, 20, 30, 40, 50} (assuming U = {1, 2, 3, ..., 1000})

To find the intersection (cap) of X' and Y, simply list the common elements between the two sets:

X' cap Y = {30, 50}

Therefore, the answer to X' cap Y is {30, 50}.

2. Y' U Z':
Similarly, we will find Y' (complement of Y) and Z':

Y' = U \ Y = {10, 20, 30, 40, 60, 70, 80, 90, 100}
Z' = U \ Z = {10, 20, 30, 50, 70, 80, 90, 100}

To find the union (U) of Y' and Z', combine the elements of the two sets without duplication:

Y' U Z' = {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}

Therefore, the answer to Y' U Z' is {10, 20, 30, 40, 50, 60, 70, 80, 90, 100}.

3. (X \ Z)' :
To find X \ Z (set difference between X and Z), exclude the elements of Z from X:

X \ Z = {10, 20, 30}
Taking the complement of (X \ Z), we exclude these elements from U:

(X \ Z)' = U \ (X \ Z) = {40, 50, 60, 70, 80, 90, 100}

Therefore, the answer to (X \ Z)' is {40, 50, 60, 70, 80, 90, 100}.

4. (Y \ Z)' U X':
To find Y \ Z (set difference between Y and Z), exclude the elements of Z from Y:

Y \ Z = {30, 50, 60, 90, 100}

Taking the complement of (Y \ Z), we exclude these elements from U:

(Y \ Z)' = U \ (Y \ Z) = {10, 20, 40, 70, 80}

To find the union of (Y \ Z)' and X', combine the elements of the two sets without duplication:

(Y \ Z)' U X' = {10, 20, 30, 40, 50, 60, 70, 80}

Therefore, the answer to (Y \ Z)' U X' is {10, 20, 30, 40, 50, 60, 70, 80}.