Question

P and Q are sunsets of the universal set U={n:1<n<10} and n is an integer. P consists of even numbers less than 7 and Q consists of prime numbers less than 10. List the elements of a)P'uQ' b)P'nQ' c)(P'nQ')' d)(PuQ)'​

Answer 1

Part A

Answer:    P' u Q' = {3,4,5,6,7,8,9}

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Work Shown:

The universal set here is U = {2,3,4,5,6,7,8,9} so we're listing all the integers between 2 and 9. We don't include 1 or 10 since n is larger than 1, and smaller than 10.

P = even numbers less than 7, subset of U

P = {2,4,6,8}

P ' = {3,5,7,9}

Note how the set P' is the set of everything in U that is not in set P. It's the opposite of set P. We call this the complement of set P.

Q = prime numbers less than 10, subset of U

Q = {2,3,5,7}

Q ' = {4,6,8,9}

Applying the union set operation on the sets P' and Q' leads to

P' u Q' = {3,5,7,9,     4,6,8,9}

All I did was combine the two sets of numbers under one umbrella. From here we toss out the duplicate entry of 9. This next step is optional, but sorting the values is standard convention.

Doing all this leads to

P' u Q' = {3,4,5,6,7,8,9}

This set represents all the values that are in set P' or in set Q' or both.

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Part B

Answer:   P' n Q' = {9}

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Work Shown:

Now we're looking at the intersection of sets P' and Q'

List out those sets

P ' = {3,5,7,9}

Q ' = {4,6,8,9}

We see that only the value 9 is in common

Therefore,

P' n Q' = {9}

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Part C

Answer:

(P' n Q')' = {2,3,4,5,6,7,8}

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Work Shown:

You start with the result from part B. Then you erase that item (9) from the universal set. Everything in this answer set is not found in the set {9}

Put another way: we're finding the complement or opposite of P' n Q'

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Part D

Answer:     (P u Q)' = {9}

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Work Shown:

P = {2,4,6,8}

Q = {2,3,5,7}

P u Q = {2,3,4,5,6,7,8}

Everything in set P u Q is either found in P, Q, or both. Any duplicates are tossed out.

Take the opposite of this to get

(P u Q)' = {9}

It is not a coincidence we get the same result as part B. It turns out that the two equations are true for any two sets P and Q

• (P u Q)' = P' n Q'
• (P n Q)' = P' u Q'

For more information, check out De Morgan's Laws. Use of a Venn Diagram may help visualize what is going on, so you can organize the values.