Simplify the expression $(A ∪ B) ∩ (A' ∪ B)$ using set identities (Commutative, Associative, Distributive, Identity, Complement, and De Morgan's Laws) to its simplest equivalent fo... Simplify the expression $(A ∪ B) ∩ (A' ∪ B)$ using set identities (Commutative, Associative, Distributive, Identity, Complement, and De Morgan's Laws) to its simplest equivalent form, and explicitly state which set identities were crucial in deriving the final answer. Read more

Steps to Solve

1. Apply the distributive law

We will apply the distributive law to the given expression $(A \cup B) \cap (A' \cup B)$. The distributive law states that $X \cap (Y \cup Z) = (X \cap Y) \cup (X \cap Z)$. Here, let $X = (A \cup B)$, $Y = A'$, and $Z = B$. Applying the distributive law, we get:

$(A \cup B) \cap (A' \cup B) = [(A \cup B) \cap A'] \cup [(A \cup B) \cap B]$

2. Apply the distributive law again

Now we apply the distributive law to each part of the expression. For the first part, $(A \cup B) \cap A' = (A \cap A') \cup (B \cap A')$. For the second part, $(A \cup B) \cap B = (A \cap B) \cup (B \cap B)$. So the expression becomes:

$[(A \cap A') \cup (B \cap A')] \cup [(A \cap B) \cup (B \cap B)]$

3. Apply the complement law

The complement law states that $A \cap A' = \emptyset$ and the identity law states that $B \cap B = B$. Substituting these into the expression:

$[\emptyset \cup (B \cap A')] \cup [(A \cap B) \cup B]$

4. Apply the identity law

The identity law states that $\emptyset \cup X = X$. Applying this, we have:

$[B \cap A'] \cup [(A \cap B) \cup B]$

5. Apply the absorption law

The absorption law can be stated as $(A \cap B) \cup B = B$. So we can simplify $(A \cap B) \cup B$ to $B$. The expression now becomes:

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

6. Apply the absorption law again

We can rewrite $(B \cap A') \cup B$ as $(A' \cap B) \cup B$. Apply the absorption law $ (X \cap Y) \cup Y = Y $, here $X = A'$ and $Y = B$, Then we have:

$B$