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    Equivalence (If and Only If / Bi-conditional)

    Rumman Ansari August 09, 2025 254 views

    🔄 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 ↔ q is true (1)

    • If both are false (0,0) → then p ↔ q is true (1)

    • But if one is true and the other is false (1,0 or 0,1) → then p ↔ q is false (0)

    đŸ”ĸ 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)