If   3, 6, n A xx n n N = =     9, 4, n B xx n n N = =   then find A B, A B and A B − .

1. Break Down the Problem

We need to find the following set operations:

1. $A \cap B$ (Intersection of sets A and B)
2. $A \cup B$ (Union of sets A and B)
3. $A - B$ (Set difference of A and B)

2. Relevant Concepts

The sets are defined as:

• $A = \{ 3, 6, n \mid n \in \mathbb{N}, n \leq 6 \}$
• $B = \{ 9, 4, n \mid n \in \mathbb{N}, n \leq 9 \}$

Calculating the elements of each set:

• Set $A$: All natural numbers $n$ where $n$ is less than or equal to 6, gives us $A = \{ 1, 2, 3, 4, 5, 6 \}$.
• Set $B$: All natural numbers $n$ where $n$ is less than or equal to 9, gives us $B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$.

3. Analysis and Detail

Now, we can perform the set operations:

3.1 Intersection $A \cap B$:

To find the elements common to both sets.

• $A \cap B = \{ 1, 2, 3, 4, 5, 6 \} \cap \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$
• Thus, $A \cap B = \{ 1, 2, 3, 4, 5, 6 \}$.

3.2 Union $A \cup B$:

To find all unique elements from both sets.

• $A \cup B = \{ 1, 2, 3, 4, 5, 6 \} \cup \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$
• Thus, $A \cup B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$.

3.3 Set Difference $A - B$:

To find elements in $A$ that are not in $B$.

• $A - B = \{ 1, 2, 3, 4, 5, 6 \} - \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$
• Thus, $A - B = \{ \}$ (the empty set).

4. Verify and Summarize

We have calculated each operation based on the definitions of the sets. The results are consistent with set theory principles.

Final Answer

1. $A \cap B = \{ 1, 2, 3, 4, 5, 6 \}$
2. $A \cup B = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$
3. $A - B = \{ \}$