We are given two sets $A$ and $B$, and we are asked to determine the cardinalities of $A$, $B$, $A \cup B$, and $A \cap B$.

Step 1: Define set $A$

Set $A$ consists of natural numbers less than 6 whose greatest common divisor (GCD) with 6 is 1. The natural numbers less than 6 are: $1, 2, 3, 4, 5$ Now, we check the GCD of each of these numbers with 6:

• $\gcd(1, 6) = 1$
• $\gcd(2, 6) = 2$
• $\gcd(3, 6) = 3$
• $\gcd(4, 6) = 2$
• $\gcd(5, 6) = 1$

So, the numbers whose GCD with 6 is 1 are $1$ and $5$. Therefore, we have: $A = \{1, 5\}$ The cardinality of $A$ is: $|A| = 2$

Step 2: Define set $B$

Set $B$ consists of divisors of 6. The divisors of 6 are: $1, 2, 3, 6$ Therefore, we have: $B = \{1, 2, 3, 6\}$ The cardinality of $B$ is: $|B| = 4$

Step 3: Find $A \cup B$ (Union of $A$ and $B$)

The union $A \cup B$ consists of all elements that are in either $A$ or $B$. Combining the elements of $A$ and $B$: $A \cup B = \{1, 5\} \cup \{1, 2, 3, 6\} = \{1, 2, 3, 5, 6\}$ The cardinality of $A \cup B$ is: $|A \cup B| = 5$

Step 4: Find $A \cap B$ (Intersection of $A$ and $B$)

The intersection $A \cap B$ consists of elements that are in both $A$ and $B$. From the sets $A = \{1, 5\}$ and $B = \{1, 2, 3, 6\}$, the common element is $1$: $A \cap B = \{1\}$ The cardinality of $A \cap B$ is: $|A \cap B| = 1$

Final Answers:

• $|A| = 2$
• $|B| = 4$
• $|A \cup B| = 5$
• $|A \cap B| = 1$

Would you like further details on any of the steps?

Here are 5 related questions to explore:

1. What is the greatest common divisor (GCD) of any two numbers?
2. How do you find the divisors of a number systematically?
3. Can you generalize the concept of GCD for different numbers like 8 or 12?
4. How do you calculate the union and intersection of sets involving more elements?
5. What are some properties of sets when considering unions and intersections?

Tip: When finding divisors of a number, always check the prime factorization—it helps in systematically listing divisors.