Logic Worksheets | Dynamic Logic Worksheets for Teachers - Free Printable

Certainly! Let’s complete each of the four truth tables step by step, explaining the logic behind each column.

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## Truth Table 1: `p → ¬a`

We need to evaluate:
- `p` (input)
- `a` (input)
- `¬a` (negation of `a`)
- `p → ¬a` (implication: if p then not a)

Truth Values for Implication (`→`):

> `P → Q` is False only when P is True and Q is False. Otherwise, it’s True.

| p | a | ¬a | p → ¬a |
|---|---|----|--------|
| T | T | F | F | ← p=T, ¬a=F → implication is F
| T | F | T | T | ← p=T, ¬a=T → implication is T
| F | T | F | T | ← p=F → implication always T
| F | F | T | T | ← p=F → implication always T

✔ Completed Table 1:

| 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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## Truth Table 2: `k ∧ ¬o`

We need to evaluate:
- `k` (input)
- `o` (input)
- `¬o` (negation of `o`)
- `k ∧ ¬o` (conjunction: k AND not o)

Truth Values for Conjunction (`∧`):

> `P ∧ Q` is True only when both P and Q are True.

| k | o | ¬o | k ∧ ¬o |
|---|---|----|--------|
| T | T | F | F | ← k=T, ¬o=F → F
| T | F | T | T | ← k=T, ¬o=T → T
| F | T | F | F | ← k=F → F
| F | F | T | F | ← k=F → F

✔ Completed Table 2:

| 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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## Truth Table 3: `¬m ∨ r`

We need to evaluate:
- `m` (input)
- `r` (input)
- `¬m` (negation of `m`)
- `¬m ∨ r` (disjunction: not m OR r)

Truth Values for Disjunction (`∨`):

> `P ∨ Q` is False only when both P and Q are False. Otherwise, True.

| m | r | ¬m | ¬m ∨ r |
|---|---|----|--------|
| T | T | F | T | ← ¬m=F, r=T → T
| T | F | F | F | ← ¬m=F, r=F → F
| F | T | T | T | ← ¬m=T → T
| F | F | T | T | ← ¬m=T → T

✔ Completed Table 3:

| m | r | ¬m | ¬m ∨ r |
|---|---|----|--------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

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## Truth Table 4: `j ↔ ¬z`

We need to evaluate:
- `j` (input)
- `z` (input)
- `¬z` (negation of `z`)
- `j ↔ ¬z` (biconditional: j if and only if not z)

Truth Values for Biconditional (`↔`):

> `P ↔ Q` is True when P and Q have the same truth value (both T or both F).

| j | z | ¬z | j ↔ ¬z |
|---|---|----|--------|
| T | T | F | F | ← j=T, ¬z=F → different → F
| T | F | T | T | ← j=T, ¬z=T → same → T
| F | T | F | T | ← j=F, ¬z=F → same → T
| F | F | T | F | ← j=F, ¬z=T → different → F

✔ Completed Table 4:

| 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 — All Completed Truth Tables:

Table 1: `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`

| 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 | r | ¬m | ¬m ∨ r |
|---|---|----|--------|
| T | T | F | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

Table 4: `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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