simplify the following in 9 steps listing each step even if the same laws are used (𝒑 ∧ (𝒑 → 𝒓 ∨ 𝒒)) ∧ (𝒓 → 𝒒)

Let's simplify the expression \((𝒑 ∧ (𝒑 → 𝒓 ∨ 𝒒)) ∧ (𝒓 → 𝒒)\) step by step using logical laws.

### Expression:
\[
(𝒑 ∧ (𝒑 → 𝒓 ∨ 𝒒)) ∧ (𝒓 → 𝒒)
\]

### Step 1: Apply **Implication Law** (\(p → q \equiv \neg p ∨ q\)) to both implications.
- \(𝒑 → 𝒓 ∨ 𝒒\) becomes \(\neg 𝒑 ∨ (𝒓 ∨ 𝒒)\).
- \(𝒓 → 𝒒\) becomes \(\neg 𝒓 ∨ 𝒒\).

Now the expression becomes:
\[
(𝒑 ∧ (\neg 𝒑 ∨ (𝒓 ∨ 𝒒))) ∧ (\neg 𝒓 ∨ 𝒒)
\]

### Step 2: Apply **Distributive Law** to the inner part \((𝒑 ∧ (\neg 𝒑 ∨ (𝒓 ∨ 𝒒)))\).
Distribute \(𝒑\) over the disjunction:
\[
(𝒑 ∧ \neg 𝒑) ∨ (𝒑 ∧ (𝒓 ∨ 𝒒))
\]
This simplifies to:
\[
𝒇 ∨ (𝒑 ∧ (𝒓 ∨ 𝒒))
\]
which simplifies further to:
\[
𝒑 ∧ (𝒓 ∨ 𝒒)
\]

### Step 3: Now substitute this back into the full expression:
\[
(𝒑 ∧ (𝒓 ∨ 𝒒)) ∧ (\neg 𝒓 ∨ 𝒒)
\]

### Step 4: Apply **Associative Law** for conjunctions.
Rearrange:
\[
𝒑 ∧ ((𝒓 ∨ 𝒒) ∧ (\neg 𝒓 ∨ 𝒒))
\]

### Step 5: Apply **Distributive Law** on \((𝒓 ∨ 𝒒) ∧ (\neg 𝒓 ∨ 𝒒)\).
Distribute \((𝒓 ∨ 𝒒)\) over \((\neg 𝒓 ∨ 𝒒)\):
\[
(𝒓 ∧ (\neg 𝒓 ∨ 𝒒)) ∨ (𝒒 ∧ (\neg 𝒓 ∨ 𝒒))
\]

### Step 6: Simplify the terms.
- \(𝒓 ∧ (\neg 𝒓 ∨ 𝒒)\) becomes \((𝒓 ∧ \neg 𝒓) ∨ (𝒓 ∧ 𝒒)\), which simplifies to \(𝒇 ∨ (𝒓 ∧ 𝒒) = 𝒓 ∧ 𝒒\).
- \(𝒒 ∧ (\neg 𝒓 ∨ 𝒒)\) simplifies to \(𝒒\).

So the whole expression becomes:
\[
(𝒓 ∧ 𝒒) ∨ 𝒒
\]

### Step 7: Apply **Absorption Law**.
\((𝒓 ∧ 𝒒) ∨ 𝒒\) simplifies to \(𝒒\).

### Step 8: Substitute back into the main expression:
\[
𝒑 ∧ 𝒒
\]

### Step 9: The simplified expression is:
\[
𝒑 ∧ 𝒒
\]

The final simplified form is \(𝒑 ∧ 𝒒\).

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Do you need more details or have any questions?

### 5 Related Questions:
1. What is the absorption law and when is it used?
2. How does the distributive law work in logic expressions?
3. Can you explain the associative law for conjunctions in more detail?
4. What is the implication law, and how does it transform implications?
5. How does simplifying a logical expression differ from simplifying an algebraic expression?

### Tip:
When simplifying logical expressions, always check for absorption or idempotent laws early on—they can drastically reduce the complexity!

This detailed solution simplifies the propositional logic expression (𝒑 ∧ (𝒑 → 𝒓 ∨ 𝒒)) ∧ (𝒓 → 𝒒) in 9 steps using key logical laws like Implication, Distributive, and Absorption Laws. Suitable for undergraduate students studying Discrete Mathematics.

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