Question

Given that n(U) =50, n(P) = 28, n(Q) = 35 and n(P'nQ') =x, Express (P'nQ), (PnQ),

and n(PnQ') in terms of x. Find the greatest and the smallest possible values of x.

Answer 1

This is a question related to set theory, a branch of mathematics that deals with the properties of well-defined collections of objects1. Here are some formulas that might help you solve the question2:

The complement of a set A is denoted by A’ and it contains all the elements that are not in A.

The union of two sets A and B is denoted by A ∪ B and it contains all the elements that are in either A or B or both.

The intersection of two sets A and B is denoted by A ∩ B and it contains all the elements that are in both A and B.

The difference of two sets A and B is denoted by A - B and it contains all the elements that are in A but not in B.

The cardinality of a set A is denoted by n(A) and it represents the number of elements in A.

Using these formulas, we can express the sets (P’nQ), (PnQ), and n(PnQ’) in terms of x as follows:

(P’nQ) = (P ∪ Q)’ = U - (P ∪ Q) = U - P - Q + P ∩ Q

(PnQ) = P ∩ Q

n(PnQ’) = n(P) + n(Q’) - n(P ∩ Q’) = n(P) + n(U) - n(Q) - n(P ∩ Q)

To find the greatest and smallest possible values of x, we need to use the following formula2:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Applying this formula to P and Q, we get:

n(P ∪ Q) = n(P) + n(Q) - n(P ∩ Q)

50 - x = 28 + 35 - n(P ∩ Q)

x + n(P ∩ Q) = 13

Since x and n(P ∩ Q) are both non-negative integers, the greatest possible value of x is 13 when n(P ∩ Q) is 0, and the smallest possible value of x is 0 when n(P ∩ Q) is 13.

I hope this helps you understand the question better.

Answer 2

Given the following information:

`\displaystyle\sf n(U) = 50`

`\displaystyle\sf n(P) = 28`

`\displaystyle\sf n(Q) = 35`

`\displaystyle\sf n(P'\cap Q') = x`

Let's express the requested values in terms of x:

1.
`\displaystyle\sf P'\cap Q`:

Using the principle of complements, we know that
`\displaystyle\sf P'\cap Q = U \setminus (P \cup Q) = U - (P \cup Q)`.

Since the universal set U has 50 elements, and we are given that
`\displaystyle\sf n(P) = 28` and
`\displaystyle\sf n(Q) = 35`, we can calculate the value of
`\displaystyle\sf n(P \cup Q)`:

`\displaystyle\sf n(P \cup Q) = n(P) + n(Q) - n(P \cap Q)`.

Since we don't have the value of
`\displaystyle\sf n(P \cap Q)`, we cannot calculate the exact value of
`\displaystyle\sf n(P \cup Q)`. Therefore, we cannot express
`\displaystyle\sf P'\cap Q` in terms of x.

2.
`\displaystyle\sf P \cap Q`:

Using the same principle of complements, we know that
`\displaystyle\sf P \cap Q = U \setminus (P' \cup Q') = U - (P' \cup Q')`.

Similarly, we cannot calculate the exact value of
`\displaystyle\sf n(P' \cup Q')` since we don't know
`\displaystyle\sf x`. Thus, we cannot express
`\displaystyle\sf P \cap Q` in terms of x.

3.
`\displaystyle\sf n(P \cap Q')`:

Using De Morgan's laws, we know that
`\displaystyle\sf n(P \cap Q') = n(P) - n(P \cap Q)`.

Since we are given that
`\displaystyle\sf n(P) = 28`, we need to calculate
`\displaystyle\sf n(P \cap Q)`. Unfortunately, we don't have enough information to find the exact value of
`\displaystyle\sf n(P \cap Q)` in terms of x. Therefore, we cannot express
`\displaystyle\sf n(P \cap Q')` in terms of x.

Regarding the greatest and smallest possible values of x, without additional information or constraints, we cannot determine those values. We would need more details or conditions related to the intersection of sets P and Q or other relevant factors to determine the range or limitations of x.