5 Truth Tables Worksheet PDF | PDF - Free Printable

Let’s solve this step by step.

We are given two truth tables to complete, and for each, we must check if a set of compound propositions forms a consistent system. A consistent system means there is at least one row in the truth table where all the given propositions are true at the same time.

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## First Truth Table

Columns: P, Q, R, P ∧ R, ~Q, Q ∨ R, P ∧ ~Q

We’ll fill in each column row by row.

Recall:
- ∧ = AND → true only if both sides are true
- ∨ = OR → true if at least one side is true
- ~ = NOT → flips T to F, F to T

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

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

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

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

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

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

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

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

Now, let’s write the full completed table:

| 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 |

Now, the question: Do the following form a consistent system?
→ P ∧ R, ~Q, Q ∨ R, P ∧ ~Q

We need to find one row where ALL FOUR are TRUE.

Check each row:

- Row 1: P∧R=T, ~Q=F → fails
- Row 2: P∧R=F → fails
- Row 3: P∧R=T, ~Q=T, Q∨R=T, P∧~Q=T → ✔ ALL TRUE!
- Row 4: P∧R=F → fails
- Rows 5–8: P∧R or P∧~Q is false → fail

✔ So YES — Row 3 makes all four true → Consistent

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## Second Truth Table

Columns: P, Q, R, P ∨ R, ~Q, P ∧ Q, ~Q ∨ R

Again, fill row by row. There are 8 rows (all combinations of T/F for P, Q, R).

We’ll go in order:

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

Row 2: P=T, Q=T, R=F

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

Row 3: P=T, Q=F, R=T

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

Row 4: P=T, Q=F, R=F

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

Row 5: P=F, Q=T, R=T

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

Row 6: P=F, Q=T, R=F

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

Row 7: P=F, Q=F, R=T

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

Row 8: P=F, Q=F, R=F

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

Completed 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, check consistency for:
→ P ∨ R, ~Q, P ∧ Q, ~Q ∨ R

Need one row where ALL FOUR are TRUE.

Check each row:

- Row 1: ~Q=F → fails
- Row 2: ~Q=F → fails
- Row 3: P∧Q=F → fails
- Row 4: P∧Q=F → fails
- Row 5: ~Q=F → fails
- Row 6: P∨R=F, ~Q=F, P∧Q=F → fails
- Row 7: P∧Q=F → fails
- Row 8: P∨R=F, P∧Q=F → fails

✘ No row has all four true → Not consistent

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Final Answer:

First system: Yes, it is consistent.
Second system: No, it is not consistent.