Let's simplify the logical expression step by step:

The given expression is:

$p \lor q \land (r \lor p) \land r$

Step 1: Apply parenthesis

First, let's look at the parentheses: $(r \lor p)$ So, the expression becomes: $p \lor q \land (r \lor p) \land r$

Step 2: Understand precedence

We need to be aware of the precedence of the logical operations:

• The AND ($\land$) operation has higher precedence than OR ($\lor$).
• So, the expression should be handled in this order:
  1. $q \land (r \lor p)$
  2. Then, $(r \lor p) \land r$
  3. Finally, apply the ORs with $p$.

Let’s break it down more clearly:

1. First, we handle $q \land (r \lor p)$, which will be: $q \land (r \lor p)$ This doesn't simplify directly unless we know the truth values of $q$, $r$, and $p$.

2. Next, simplify $(r \lor p) \land r$. This can be simplified using the distributive property: $(r \lor p) \land r = r$ since $r \land r = r$ and $p \land r = r$ (because $r$ is present in both).

Step 3: Final simplification

Now, we substitute this back into the original expression: $p \lor (q \land r)$

Final Expression:

Thus, the simplified expression is: $p \lor (q \land r)$

This is the most simplified version of the expression.

Questions:

1. What would happen if you had a NOT operation in this expression?
2. How do you simplify expressions with more than three variables?
3. Can this simplification be proven by truth tables?
4. How would the simplification change if $q$ was false?
5. How does associativity affect logical expressions?

Tip:

Always pay attention to the precedence of logical operators when simplifying expressions. $\land$ (AND) always takes precedence over $\lor$ (OR), so break down the expressions accordingly.

Let me know if you need further clarification or if you have any other questions!