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 need to check if a set of compound propositions form 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 look at the question:

> Do the following compound propositions 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: T, F, T, F → no
- Row 2: F, F, T, F → no
- Row 3: T, T, T, T → ✔ YES! All four are true here.
- Row 4: F, T, F, T → no (Q∨R is false)
- Rows 5–8: all have at least one false

So yes — Row 3 makes all four true → consistent system

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## Second Truth Table

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

Again, fill row by row. We’ll use same logic.

There are 8 rows (all combinations of T/F for P, Q, R). Let’s list them in order:

Row 1: P=T, Q=T, R=T
Row 2: P=T, Q=T, R=F
Row 3: P=T, Q=F, R=T
Row 4: P=T, Q=F, R=F
Row 5: P=F, Q=T, R=T
Row 6: P=F, Q=T, R=F
Row 7: P=F, Q=F, R=T
Row 8: P=F, Q=F, R=F

Now compute each column:

Row 1: T,T,T

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

Row 2: T,T,F

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

Row 3: T,F,T

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

Row 4: T,F,F

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

Row 5: F,T,T

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

Row 6: F,T,F

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

Row 7: F,F,T

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

Row 8: F,F,F

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

Now, the question:

> Do the following compound propositions form a consistent system?
> P ∨ R, ~Q, P ∧ Q, ~Q ∨ R

We need one row where ALL FOUR are TRUE.

Check each row:

- Row 1: T, F, T, T → ~Q is false → no
- Row 2: T, F, T, F → ~Q and last are false → no
- Row 3: T, T, F, T → P∧Q is false → no
- Row 4: T, T, F, T → P∧Q is false → no
- Row 5: T, F, F, T → ~Q and P∧Q false → no
- Row 6: F, F, F, F → all bad → no
- Row 7: T, T, F, T → P∧Q is false → no
- Row 8: F, T, F, T → P∨R and P∧Q false → no

Wait — is there ANY row where all four are true?

Look again:

We need:
- P ∨ R = T
- ~Q = T → so Q must be F
- P ∧ Q = T → but if Q is F, then P ∧ Q = F → impossible!

Ah! Here’s the key:

If ~Q is true → Q is false → then P ∧ Q = P ∧ false = false → can NEVER be true.

So it’s impossible for both ~Q and P ∧ Q to be true at the same time.

Therefore, no row can make all four true → not consistent

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Final Answer:

First system: Yes, they form a consistent system.
Second system: No, they do not form a consistent system.