The sets P = {x: x is a prime factor of 42} and Q = {x: x is a factor of 24} are subsets of the U = {x: x is an integer}. List the elements of (a) P ∩ Q, (b) P ∪ Q.

Understand the Problem

The question is asking to identify the elements of two sets, P and Q. Specifically, we need to find (a) the intersection of set P and set Q, and (b) the union of set P and set Q. Set P consists of the prime factors of 42, while set Q consists of the factors of 24.

Answer

(a) $ P \cap Q = \{2, 3\} $ (b) $ P \cup Q = \{1, 2, 3, 4, 6, 7, 8, 12, 24\} $

Steps to Solve

Identify Set P
Set P consists of the prime factors of 42.
To find the prime factors, we can factor 42:
$$ 42 = 2 \times 3 \times 7 $$
Thus, the prime factors are:
$$ P = {2, 3, 7} $$
Identify Set Q
Set Q consists of the factors of 24.
To find the factors, we can list all integers that divide 24 without a remainder:
$$ Q = {1, 2, 3, 4, 6, 8, 12, 24} $$
Find the Intersection (P ∩ Q)
The intersection of the two sets includes elements that are in both P and Q:
$$ P \cap Q = {x: x \in P \text{ and } x \in Q} $$
Looking at both sets:
$$ P \cap Q = {2, 3} $$
Find the Union (P ∪ Q)
The union of the two sets includes all unique elements from both sets:
$$ P \cup Q = {x: x \in P \text{ or } x \in Q} $$
Combining the elements:
$$ P \cup Q = {1, 2, 3, 4, 6, 7, 8, 12, 24} $$

(a) ( P \cap Q = {2, 3} )

(b) ( P \cup Q = {1, 2, 3, 4, 6, 7, 8, 12, 24} )

More Information

This problem combines basic factorization and set theory concepts. Understanding the distinction between prime factors and factors is essential, as well as how intersection and union work in set relations.

Tips

Confusing prime factors with all factors: Remember, prime factors are only prime numbers that multiply together to equal the original number.
Not listing unique elements when finding a union: Ensure that each element appears only once in the union.

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