Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7\}$, and sets $A = \{1, 3, 5\}$, $B = \{2, 4, 6\}$, and $C = \{3, 4, 5, 6, 7\}$. We need to verify the following set identities: a) $(A \cup B)' = A' \cap B'$ b) $(B \cap C)' = B' \cup C'$ c) $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$ d) $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$

1. Problem Description

Given the universal set $U = \{1, 2, 3, 4, 5, 6, 7\}$, and sets $A = \{1, 3, 5\}$, $B = \{2, 4, 6\}$, and $C = \{3, 4, 5, 6, 7\}$. We need to verify the following set identities:

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

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

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

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

2. Solution Steps

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

First, let's calculate $A \cup B$:

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

Now, let's find $(A \cup B)'$:

$(A \cup B)' = U - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7\} - \{1, 2, 3, 4, 5, 6\} = \{7\}$

Next, let's calculate $A'$ and $B'$:

$A' = U - A = \{1, 2, 3, 4, 5, 6, 7\} - \{1, 3, 5\} = \{2, 4, 6, 7\}$

$B' = U - B = \{1, 2, 3, 4, 5, 6, 7\} - \{2, 4, 6\} = \{1, 3, 5, 7\}$

Now, let's find $A' \cap B'$:

$A' \cap B' = \{2, 4, 6, 7\} \cap \{1, 3, 5, 7\} = \{7\}$

Since $(A \cup B)' = \{7\}$ and $A' \cap B' = \{7\}$, the identity $(A \cup B)' = A' \cap B'$ holds true.

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

First, let's calculate $B \cap C$:

$B \cap C = \{2, 4, 6\} \cap \{3, 4, 5, 6, 7\} = \{4, 6\}$

Now, let's find $(B \cap C)'$:

$(B \cap C)' = U - (B \cap C) = \{1, 2, 3, 4, 5, 6, 7\} - \{4, 6\} = \{1, 2, 3, 5, 7\}$

Next, let's calculate $B'$ and $C'$:

$B' = U - B = \{1, 2, 3, 4, 5, 6, 7\} - \{2, 4, 6\} = \{1, 3, 5, 7\}$

$C' = U - C = \{1, 2, 3, 4, 5, 6, 7\} - \{3, 4, 5, 6, 7\} = \{1, 2\}$

Now, let's find $B' \cup C'$:

$B' \cup C' = \{1, 3, 5, 7\} \cup \{1, 2\} = \{1, 2, 3, 5, 7\}$

Since $(B \cap C)' = \{1, 2, 3, 5, 7\}$ and $B' \cup C' = \{1, 2, 3, 5, 7\}$, the identity $(B \cap C)' = B' \cup C'$ holds true.

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

We already have $A \cup B = \{1, 2, 3, 4, 5, 6\}$ from part (a).

$(A \cup B) \cap C = \{1, 2, 3, 4, 5, 6\} \cap \{3, 4, 5, 6, 7\} = \{3, 4, 5, 6\}$

Now let's calculate $A \cap C$ and $B \cap C$:

$A \cap C = \{1, 3, 5\} \cap \{3, 4, 5, 6, 7\} = \{3, 5\}$

$B \cap C = \{2, 4, 6\} \cap \{3, 4, 5, 6, 7\} = \{4, 6\}$

Now, let's find $(A \cap C) \cup (B \cap C)$:

$(A \cap C) \cup (B \cap C) = \{3, 5\} \cup \{4, 6\} = \{3, 4, 5, 6\}$

Since $(A \cup B) \cap C = \{3, 4, 5, 6\}$ and $(A \cap C) \cup (B \cap C) = \{3, 4, 5, 6\}$, the identity $(A \cup B) \cap C = (A \cap C) \cup (B \cap C)$ holds true.

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

First, let's calculate $A \cap B$:

$A \cap B = \{1, 3, 5\} \cap \{2, 4, 6\} = \{\}$ (empty set)

$(A \cap B) \cup C = \{\} \cup \{3, 4, 5, 6, 7\} = \{3, 4, 5, 6, 7\}$

Next, let's calculate $A \cup C$ and $B \cup C$:

$A \cup C = \{1, 3, 5\} \cup \{3, 4, 5, 6, 7\} = \{1, 3, 4, 5, 6, 7\}$

$B \cup C = \{2, 4, 6\} \cup \{3, 4, 5, 6, 7\} = \{2, 3, 4, 5, 6, 7\}$

Now, let's find $(A \cup C) \cap (B \cup C)$:

$(A \cup C) \cap (B \cup C) = \{1, 3, 4, 5, 6, 7\} \cap \{2, 3, 4, 5, 6, 7\} = \{3, 4, 5, 6, 7\}$

Since $(A \cap B) \cup C = \{3, 4, 5, 6, 7\}$ and $(A \cup C) \cap (B \cup C) = \{3, 4, 5, 6, 7\}$, the identity $(A \cap B) \cup C = (A \cup C) \cap (B \cup C)$ holds true.

3. Final Answer

All four identities hold true:

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

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

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

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