Let's solve both parts step by step:

Given Sets:

• $A = \{1, 2, 3, 4, 5\}$
• $B = \{2, 3, 6, 8\}$

a) $B \times A$ (Cartesian Product of $B$ and $A$)

The Cartesian product $B \times A$ is the set of all ordered pairs $(b, a)$, where $b \in B$ and $a \in A$.

So, we take every element from set $B$ and pair it with every element from set $A$.

$B \times A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (8, 1), (8, 2), (8, 3), (8, 4), (8, 5)\}$

b) $A \Delta B$ (Symmetric Difference of $A$ and $B$)

The symmetric difference $A \Delta B$ is the set of elements that are in $A$ or $B$ but not in both. In other words, it is the union of $A \setminus B$ and $B \setminus A$.

• $A \setminus B = \{1, 4, 5\}$ (elements in $A$ but not in $B$)
• $B \setminus A = \{6, 8\}$ (elements in $B$ but not in $A$)

So, $A \Delta B = \{1, 4, 5, 6, 8\}$.

Final Answers:

a) $B \times A = \{(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (8, 1), (8, 2), (8, 3), (8, 4), (8, 5)\}$

b) $A \Delta B = \{1, 4, 5, 6, 8\}$

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Would you like more details or have any questions on this?

Here are 5 related questions for further practice:

1. Find the Cartesian product $A \times B$.
2. Determine $(A \cap B)$, the intersection of sets $A$ and $B$.
3. Find $A \cup B$, the union of sets $A$ and $B$.
4. What is the complement of set $A$ if the universal set is $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$?
5. Find the symmetric difference $(A \Delta B) \times B$.

Tip: The symmetric difference can be viewed as the set of elements that are "exclusive" to each set, meaning they belong to one set or the other, but not both.