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Without using truth table show that the following proportion are logically equivalent

The expression given is:

\[
(p \land q) \lor (p \lor q) \equiv p
\]

To show this equivalence without a truth table, we can use logical identities. Here's a step-by-step explanation:

1. **Distributive Property**: We start with the expression on the left side:  
   \[
   (p \land q) \lor (p \lor q)
   \]
   We can distribute \((p \lor q)\) over \((p \land q)\):
   \[
   = (p \lor q) \lor (p \land q)
   \]

2. **Absorption Law**: According to the absorption law in logic, \( p \lor (p \land q) \equiv p \). Applying this to our expression:
   \[
   = p
   \]

Thus, we have shown that:
\[
(p \land q) \lor (p \lor q) \equiv p
\]

Therefore, the expression is logically equivalent to \( p \) without using a truth table.

Would you like more details on the logic laws applied here or have any questions?

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**Related Questions:**

1. What are other methods to prove logical equivalences?
2. Can you show an example where distributive property is applied in logical expressions?
3. How does the absorption law work in logical equivalences?
4. Are there any cases where distributive property does not simplify an expression?
5. How does the commutative property affect logical expressions?

**Tip:** Remember that understanding the basic logical identities, like distributive, absorption, and De Morgan's laws, can simplify complex logical expressions.

Without using truth table show that the following proposition is logically equivalent: \((p \land q) \lor (p \lor q) \equiv p\)

Learn how to show the logical equivalence of (p ∧ q) ∨ (p ∨ q) ≡ p without using a truth table. This solution uses key logical identities such as the distributive property and absorption law. Suitable for high school students in Grades 10-12.

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