6 Truth Tables Worksheet Solutions PDF | PDF - Free Printable

Let's carefully analyze and solve the logic problems presented in your image, including completing truth tables and determining whether the compound propositions form a consistent system.

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🔹 Problem 1: Complete the Truth Table

We are given a partial truth table with columns:

| P | Q | R | P ∧ R | ~Q | Q ∨ R | P ∧ ~Q |

And we need to complete it for all 8 combinations of P, Q, R (since there are 3 variables, 2³ = 8 rows).

Let’s go row by row.

#### Row 1: P=T, Q=T, R=T
- P ∧ R = T ∧ T = T
- ~Q = ~T = F
- Q ∨ R = T ∨ T = T
- P ∧ ~Q = T ∧ F = F

✔ Already filled correctly.

#### Row 2: P=T, Q=T, R=F
- P ∧ R = T ∧ F = F
- ~Q = ~T = F
- Q ∨ R = T ∨ F = T
- P ∧ ~Q = T ∧ F = F

✔ Correct.

#### Row 3: P=T, Q=F, R=T
- P ∧ R = T ∧ T = T
- ~Q = ~F = T
- Q ∨ R = F ∨ T = T
- P ∧ ~Q = T ∧ T = T

✔ Correct.

#### Row 4: P=T, Q=F, R=F
- P ∧ R = T ∧ F = F
- ~Q = ~F = T
- Q ∨ R = F ∨ F = F
- P ∧ ~Q = T ∧ T = T

So far:
| P | Q | R | P∧R | ~Q | Q∨R | P∧~Q |
|---|---|---|-----|----|-----|------|
| T | T | T | T | F | T | F |
| T | T | F | F | F | T | F |
| T | F | T | T | T | T | T |
| T | F | F | F | T | F | T |

Wait — the table says Q∨R is T here, but Q=F, R=F → Q∨R = F, so this entry is incorrect.

But looking at the original image, the user has already filled some values. Let's check:

> In the third row: Q∨R = T → But Q=F, R=T → F∨T = T → ✔ OK
> In fourth row: Q∨R = F → Q=F, R=F → F∨F = F → ✔ OK

So let's continue.

#### Row 5: P=F, Q=T, R=T
- P ∧ R = F ∧ T = F
- ~Q = ~T = F
- Q ∨ R = T ∨ T = T
- P ∧ ~Q = F ∧ F = F

✔ Matches.

#### Row 6: P=F, Q=T, R=F
- P ∧ R = F ∧ F = F
- ~Q = ~T = F
- Q ∨ R = T ∨ F = T
- P ∧ ~Q = F ∧ F = F

✔ Matches.

#### Row 7: P=F, Q=F, R=T
- P ∧ R = F ∧ T = F
- ~Q = ~F = T
- Q ∨ R = F ∨ T = T
- P ∧ ~Q = F ∧ T = F

✔ Matches.

#### Row 8: P=F, Q=F, R=F
- P ∧ R = F ∧ F = F
- ~Q = ~F = T
- Q ∨ R = F ∨ F = F
- P ∧ ~Q = F ∧ T = F

✔ All correct.

So the completed truth table is:

| P | Q | R | P∧R | ~Q | Q∨R | P∧~Q |
|---|---|---|-----|----|-----|------|
| T | T | T | T | F | T | F |
| T | T | F | F | F | T | F |
| T | F | T | T | T | T | T |
| T | F | F | F | T | F | T |
| F | T | T | F | F | T | F |
| F | T | F | F | F | T | F |
| F | F | T | F | T | T | F |
| F | F | F | F | T | F | F |

✔ Fully completed.

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🔹 Question 1: Do the following compound propositions form a consistent system?

Propositions:
1. $ P \land R $
2. $ \neg Q $
3. $ Q \lor R $
4. $ P \land \neg Q $

We want to know if there exists at least one row where all four are true simultaneously.

Let’s check each row:

- Row 1: P=T, Q=T, R=T
- $ P \land R $ = T
- $ \neg Q $ = F ✘
→ Not all true

- Row 2: P=T, Q=T, R=F
- $ P \land R $ = F ✘

- Row 3: P=T, Q=F, R=T
- $ P \land R $ = T
- $ \neg Q $ = T
- $ Q \lor R $ = F ∨ T = T
- $ P \land \neg Q $ = T ∧ T = T
→ All four are T ✔

So yes, row 3 satisfies all four statements.

👉 Answer: Yes, due to the third line of the truth table. ✔️

Correct!

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🔹 Problem 2: Complete the second truth table

Columns:
P, Q, R, P∨R, ~Q, P∧Q, ~Q∨R

Let’s compute each column.

#### Row 1: P=T, Q=T, R=T
- P∨R = T∨T = T
- ~Q = ~T = F
- P∧Q = T∧T = T
- ~Q∨R = F∨T = T

✔ Given as: T, F, T, T → Correct

#### Row 2: P=T, Q=T, R=F
- P∨R = T∨F = T
- ~Q = F
- P∧Q = T
- ~Q∨R = F∨F = F

Given: T, F, T, F → ✔

#### Row 3: P=T, Q=F, R=T
- P∨R = T∨T = T
- ~Q = T
- P∧Q = T∧F = F
- ~Q∨R = T∨T = T

Given: T, T, F, T → ✔

#### Row 4: P=T, Q=F, R=F
- P∨R = T∨F = T
- ~Q = T
- P∧Q = T∧F = F
- ~Q∨R = T∨F = T

Given: T, T, F, T → ✔

#### Row 5: P=F, Q=T, R=T
- P∨R = F∨T = T
- ~Q = F
- P∧Q = F∧T = F
- ~Q∨R = F∨T = T

Given: T, F, F, T → ✔

#### Row 6: P=F, Q=T, R=F
- P∨R = F∨F = F
- ~Q = F
- P∧Q = F∧T = F
- ~Q∨R = F∨F = F

Given: T, F, F, F → Wait! P∨R = F, not T → Error in the table?

But the table says P∨R = T → ✘ Incorrect

Actually, P=F, R=F → P∨R = F

So this row should be:
- P∨R = F
- ~Q = F
- P∧Q = F
- ~Q∨R = F∨F = F

So the row should be: F, F, F, F

But the table says P∨R = T → Incorrect

Let’s recheck the image:

> Row 6: P=F, Q=T, R=F → P∨R = T → This is wrong.

P∨R = F∨F = F → So this entry is incorrect.

Similarly, Row 7: P=F, Q=F, R=T
- P∨R = F∨T = T
- ~Q = T
- P∧Q = F∧F = F
- ~Q∨R = T∨T = T

Given: F, T, F, T → ✔

Row 8: P=F, Q=F, R=F
- P∨R = F∨F = F
- ~Q = T
- P∧Q = F
- ~Q∨R = T∨F = T

Given: F, T, F, T → ✔

So only Row 6 has an error: P∨R should be F, not T.

Let’s fix the full table:

| P | Q | R | P∨R | ~Q | P∧Q | ~Q∨R |
|---|---|---|-----|----|-----|------|
| T | T | T | T | F | T | T |
| T | T | F | T | F | T | F |
| T | F | T | T | T | F | T |
| T | F | F | T | T | F | T |
| F | T | T | T | F | F | T |
| F | T | F | F | F | F | F |
| F | F | T | T | T | F | T |
| F | F | F | F | T | F | T |

✔ Now corrected.

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🔹 Question 2: Do these propositions form a consistent system?

Propositions:
1. $ P \lor R $
2. $ \neg Q $
3. $ P \land Q $
4. $ \neg Q \lor R $

We need to find a row where all four are true.

Let’s check each row:

- Row 1: P=T, Q=T, R=T
- $ P \lor R $ = T
- $ \neg Q $ = F ✘
→ No

- Row 2: P=T, Q=T, R=F
- $ P \lor R $ = T
- $ \neg Q $ = F ✘

- Row 3: P=T, Q=F, R=T
- $ P \lor R $ = T
- $ \neg Q $ = T
- $ P \land Q $ = T ∧ F = F ✘
→ No

- Row 4: P=T, Q=F, R=F
- $ P \lor R $ = T
- $ \neg Q $ = T
- $ P \land Q $ = T ∧ F = F ✘
→ No

- Row 5: P=F, Q=T, R=T
- $ P \lor R $ = T
- $ \neg Q $ = F ✘

- Row 6: P=F, Q=T, R=F
- $ P \lor R $ = F ✘
→ No

- Row 7: P=F, Q=F, R=T
- $ P \lor R $ = T
- $ \neg Q $ = T
- $ P \land Q $ = F ∧ F = F ✘
→ No

- Row 8: P=F, Q=F, R=F
- $ P \lor R $ = F ✘
→ No

So no row has all four statements true.

Therefore, no consistent assignment makes all propositions true.

👉 Answer: No, since there is no line in the truth table where all of the statements are true at the same time. ✔️

Correct!

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✔ Final Summary

#### Truth Table 1 (Completed):
All entries are correct as per the image (except possibly a typo in Q∨R in row 4 — but actually it was correct: F∨F = F).

#### Consistency Check 1:
- Yes, because row 3 satisfies all four:
$ P \land R = T $, $ \neg Q = T $, $ Q \lor R = T $, $ P \land \neg Q = T $

#### Truth Table 2 (Corrected):
- Row 6: P∨R should be F, not T (as originally shown). Otherwise, all others are correct.

#### Consistency Check 2:
- No, because no row has all four propositions true simultaneously.

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🟩 Final Answers:

1. First system: Yes, consistent (due to row 3).
2. Second system: No, inconsistent (no row satisfies all).

✔ Your answers are correct!

Just note: The second truth table has an error in Row 6, where P∨R is incorrectly listed as T when it should be F. That might be a typo in the original document.

Let me know if you'd like a clean version of both tables!