To determine if two statements are equivalent, you typically construct a truth table or use logical equivalences to manipulate the statements until they match.
Here's a breakdown of methods to find equivalent statements:
1. Using Truth Tables
The most reliable method is to construct a truth table for each statement. Two statements, p and q, are logically equivalent if and only if their truth values are identical for every possible combination of truth values of their components.
- Construct the truth table: Create a table with columns for each simple statement involved (e.g., p, q) and their possible truth values (True/False or T/F). Also create columns for the compound statements you are comparing.
- Evaluate truth values: Fill in the truth values for each compound statement based on the truth values of its components and the logical operators involved (e.g., AND, OR, NOT, IMPLIES, BICONDITIONAL).
- Compare the columns: If the columns corresponding to the two statements you are comparing are identical, then the statements are logically equivalent.
Example:
Let's check if p → q (p implies q) is equivalent to ¬p ∨ q (not p or q).
| p | q | p → q | ¬p | ¬p ∨ q |
|---|---|---|---|---|
| True | True | True | False | True |
| True | False | False | False | False |
| False | True | True | True | True |
| False | False | True | True | True |
Since the columns for p → q and ¬p ∨ q are identical, the two statements are logically equivalent.
2. Using Logical Equivalences (Laws of Logic)
Another approach is to use known logical equivalences to transform one statement into the other. Here are some common logical equivalences:
- Commutative Laws:
- p ∧ q ≡ q ∧ p
- p ∨ q ≡ q ∨ p
- Associative Laws:
- (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
- (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
- Distributive Laws:
- p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
- p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
- Identity Laws:
- p ∧ True ≡ p
- p ∨ False ≡ p
- Negation Laws:
- p ∧ ¬p ≡ False
- p ∨ ¬p ≡ True
- Double Negation Law:
- ¬(¬p) ≡ p
- De Morgan's Laws:
- ¬(p ∧ q) ≡ ¬p ∨ ¬q
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
- Conditional Law:
- p → q ≡ ¬p ∨ q
- Biconditional Law:
- p ↔ q ≡ (p → q) ∧ (q → p)
Example:
Let's prove the same equivalence as above, p → q ≡ ¬p ∨ q, using logical equivalences.
We can directly apply the Conditional Law:
- p → q ≡ ¬p ∨ q
Therefore, the two statements are equivalent.
3. Using a Biconditional Statement
You can create a biconditional statement (p ↔ q), which means "p if and only if q". Then, create a truth table for p ↔ q. If the resulting truth table contains only True values (i.e., it's a tautology), then p and q are logically equivalent. If there is even one False value, the statements are not equivalent.
Example:
To show p → q ≡ ¬p ∨ q, construct the truth table for (p → q) ↔ (¬p ∨ q):
| p | q | p → q | ¬p | ¬p ∨ q | (p → q) ↔ (¬p ∨ q) |
|---|---|---|---|---|---|
| True | True | True | False | True | True |
| True | False | False | False | False | True |
| False | True | True | True | True | True |
| False | False | True | True | True | True |
Since the last column is all True, the biconditional is a tautology, and therefore p → q and ¬p ∨ q are logically equivalent.
In summary, finding equivalent statements involves verifying that they always have the same truth value through truth tables, manipulation using logical equivalences, or by showing that their biconditional statement is a tautology.
