Logic Worksheets | Dynamic Logic Worksheets for Teachers - Free Printable

Let's solve each of the four truth tables step by step. We'll fill in all possible combinations of truth values for the variables and compute the results based on logical operations.

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1. Truth Table: `p`, `a`, `~a`, `p -> ~a`

We have two variables: `p` and `a`. So there are 4 combinations.

| p | a | ~a | p -> ~a |
|---|---|----|--------|
| T | T | F | T |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

Explanation:
- `~a` is the negation of `a`: if `a` is True, `~a` is False, and vice versa.
- `p -> ~a` is a conditional (implication): It's False only when `p` is True and `~a` is False. Otherwise, it's True.

Check each row:
1. p=T, a=T → ~a=F → T→F = F? Wait! Let's recalculate carefully.

Wait — I made a mistake above. Let me correct:

For p → ~a, we need to evaluate:
- If `p` is True, then `~a` must be True for the implication to be True.
- If `p` is False, the implication is always True.

So let’s go row by row:

| p | a | ~a | p → ~a |
|---|---|----|--------|
| T | T | F | T → F = F |
| T | F | T | T → T = T |
| F | T | F | F → F = T |
| F | F | T | F → T = T |

✔ Corrected table:

| p | a | ~a | p → ~a |
|---|---|----|--------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

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2. Truth Table: `k`, `o`, `~o`, `k ^ ~o`

This is a conjunction (`^` means AND).

| k | o | ~o | k ^ ~o |
|---|---|----|--------|
| T | T | F | T ^ F = F |
| T | F | T | T ^ T = T |
| F | T | F | F ^ F = F |
| F | F | T | F ^ T = F |

So:

| k | o | ~o | k ^ ~o |
|---|---|----|--------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |

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3. Truth Table: `m`, `r`, `~m`, `~m V r`

This is a disjunction (`V` means OR). Remember: OR is True if at least one is True.

| m | r | ~m | ~m V r |
|---|---|----|--------|
| T | T | F | F V T = T |
| T | F | F | F V F = F |
| F | T | T | T V T = T |
| F | F | T | T V F = T |

So:

| m | r | ~m | ~m V r |
|---|---|----|--------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

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4. Truth Table: `j`, `z`, `~z`, `j <-> ~z`

This is a biconditional (`<->`), which is True when both sides have the same truth value.

| j | z | ~z | j <-> ~z |
|---|---|----|----------|
| T | T | F | T <-> F = F |
| T | F | T | T <-> T = T |
| F | T | F | F <-> F = T |
| F | F | T | F <-> T = F |

So:

| j | z | ~z | j <-> ~z |
|---|---|----|----------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | T |
| F | F | T | F |

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✔ Final Answer: Completed Truth Tables

#### Table 1: p, a, ~a, p -> ~a

| p | a | ~a | p -> ~a |
|---|---|----|--------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

#### Table 2: k, o, ~o, k ^ ~o

| k | o | ~o | k ^ ~o |
|---|---|----|--------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |

#### Table 3: m, r, ~m, ~m V r

| m | r | ~m | ~m V r |
|---|---|----|--------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

#### Table 4: j, z, ~z, j <-> ~z

| j | z | ~z | j <-> ~z |
|---|---|----|----------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | T |
| F | F | T | F |

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📌 Summary of Logic Rules Used:

- Negation (~): Flips the truth value.
- Implication (→): `p → q` is False only when p is True and q is False.
- Conjunction (^): True only if both operands are True.
- Disjunction (V): True if at least one operand is True.
- Biconditional (<->): True when both sides are equal.

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