Equivalence (If and Only If / Bi-conditional)
☰Fullscreen
Table of Content:
🔄 Equivalence (If and Only If / Bi-conditional)
In logic, an equivalence or bi-conditional statement is written as:
👉 p ↔ q
Read as: "p if and only if q"
🧠 What does it mean?
A bi-conditional statement is true (1) when both p and q have the same value.
-
If both are true (1,1) → then
p ↔ qis true (1) -
If both are false (0,0) → then
p ↔ qis true (1) -
But if one is true and the other is false (1,0 or 0,1) → then
p ↔ qis false (0)
🔢 Truth Table for p ↔ q (Using 0 and 1)
| p | q | p ↔ q |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
📌 Summary:
| Condition | Result |
|---|---|
| Both same (0,0 or 1,1) | ✅ 1 (True) |
| One true, one false (0,1 or 1,0) | ❌ 0 (False) |
Scenario: Door Lock with Two Keys
-
p = "Key A is turned ON"
-
q = "Key B is turned ON"
-
The door will open if and only if both keys are in the same position (either both ON or both OFF).
Truth Table with Example:
| p (Key A) | q (Key B) | p ↔ q | Door Status |
|---|---|---|---|
| 0 (OFF) | 0 (OFF) | 1 | ✅ Door Opens (same) |
| 0 (OFF) | 1 (ON) | 0 | ❌ Door Locked (different) |
| 1 (ON) | 0 (OFF) | 0 | ❌ Door Locked (different) |
| 1 (ON) | 1 (ON) | 1 | ✅ Door Opens (same) |