Let's solve the given problems step by step.

Problem 15: Matrix Subtraction $A - B$

We have matrices:

$A = \begin{bmatrix} 1 & 3 & 4 \\ 2 & 1 & 6 \end{bmatrix}$ $B = \begin{bmatrix} 2 & 2 & 1 \\ 1 & 3 & -2 \end{bmatrix}$

Solution for $A - B$:

Subtract corresponding elements:

$A - B = \begin{bmatrix} 1 - 2 & 3 - 2 & 4 - 1 \\ 2 - 1 & 1 - 3 & 6 - (-2) \end{bmatrix}$

$= \begin{bmatrix} -1 & 1 & 3 \\ 1 & -2 & 8 \end{bmatrix}$

So,
$A - B = \begin{bmatrix} -1 & 1 & 3 \\ 1 & -2 & 8 \end{bmatrix}$

Problem 16: Set Identities Verification

Given sets:
$A = \{a, b, c, d, e, f\}, \quad B = \{c, d, e\}, \quad C = \{d, e, f, g\}$

We need to prove:

(a) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

1. Find $B \cap C$: Elements common in $B$ and $C$.
   $B \cap C = \{c, d, e\} \cap \{d, e, f, g\} = \{d, e\}$

2. Compute $A \cup (B \cap C)$:
   $A \cup \{d, e\} = \{a, b, c, d, e, f\}$

3. Compute $A \cup B$:
   $A \cup B = \{a, b, c, d, e, f\}$

4. Compute $A \cup C$:
   $A \cup C = \{a, b, c, d, e, f, g\}$

5. Compute $(A \cup B) \cap (A \cup C)$:
   $\{a, b, c, d, e, f\} \cap \{a, b, c, d, e, f, g\} = \{a, b, c, d, e, f\}$

Since both sides are equal, the identity is proved.

(b) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

1. Compute $B \cup C$:
   $B \cup C = \{c, d, e\} \cup \{d, e, f, g\} = \{c, d, e, f, g\}$

2. Compute $A \cap (B \cup C)$:
   $\{a, b, c, d, e, f\} \cap \{c, d, e, f, g\} = \{c, d, e, f\}$

3. Compute $A \cap B$:
   $\{a, b, c, d, e, f\} \cap \{c, d, e\} = \{c, d, e\}$

4. Compute $A \cap C$:
   $\{a, b, c, d, e, f\} \cap \{d, e, f, g\} = \{d, e, f\}$

5. Compute $(A \cap B) \cup (A \cap C)$:
   $\{c, d, e\} \cup \{d, e, f\} = \{c, d, e, f\}$

Since both sides are equal, the identity is proved.

Final Answers:

1. $A - B = \begin{bmatrix} -1 & 1 & 3 \\ 1 & -2 & 8 \end{bmatrix}$
2. The given set identities are proved.

Would you like further explanation or details?

Related Questions:

1. How do you find the union and intersection of sets?
2. What is the general formula for matrix subtraction?
3. How can Venn diagrams help visualize set operations?
4. What are some real-life applications of set theory?
5. How do we verify set identities algebraically?

Tip:

When verifying set identities, always compute both sides separately and compare the results!