Logic worksheet | PDF - Free Printable

Let’s go step by step to complete the truth tables.

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Part (a): Truth table for conjunction (“and”) — P ∧ Q

Conjunction means “both must be true” for the result to be true.
So, P ∧ Q is true only when both P and Q are true. In all other cases, it’s false.

We fill in the last column based on that rule:

- When P = T, Q = T → P ∧ Q = T
- When P = T, Q = F → P ∧ Q = F
- When P = F, Q = T → P ∧ Q = F
- When P = F, Q = F → P ∧ Q = F

✔ So the completed table for (a) is:

| P | Q | P ∧ Q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

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Part (b): Truth table for biconditional (“if and only if”) — P ↔ Q

Biconditional means “P and Q have the same truth value.”
It’s true when both are true OR both are false.
It’s false when one is true and the other is false.

So:

- When P = T, Q = T → same → P ↔ Q = T
- When P = T, Q = F → different → P ↔ Q = F
- When P = F, Q = T → different → P ↔ Q = F
- When P = F, Q = F → same → P ↔ Q = T

✔ So the completed table for (b) is:

| P | Q | P ↔ Q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

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Why do we say the biconditional defines “logical equivalence”?

Because two statements are logically equivalent if they always have the same truth value — no matter what.
The biconditional P ↔ Q is true exactly when P and Q match in truth value.
So if P ↔ Q is always true (in every row), then P and Q are logically equivalent.

In short:
→ If P ↔ Q is a tautology (always true), then P and Q mean the same thing logically.

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Final Answer:

(a) Conjunction (P ∧ Q):

| P | Q | P ∧ Q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

(b) Biconditional (P ↔ Q):

| P | Q | P ↔ Q |
|---|---|-------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |

The biconditional defines logical equivalence because it is true precisely when both statements have the same truth value — meaning they behave identically in logic.