Which of the following connectives satisfy commutatiive law ?

Options

A. ` ^^ `
B. ` vv`
C. ` harr`
D. `all the above

Answer

To determine which of the given connectives satisfy the commutative law, we will analyze the AND operator, OR operator, and Biconditional operator using truth tables.

### Step-by-Step Solution:

1. **Understanding Commutative Law**:
The commutative law states that for two statements A and B, the operation O satisfies the law if:
\[ A \, O \, B = B \, O \, A \]
This means the order of the operands does not affect the result of the operation.

2. **Constructing Truth Tables**:
We will create truth tables for each operator: AND, OR, and Biconditional.

3. **Truth Table for AND Operator**:
- The AND operator (denoted as ∧) is true only when both operands are true.
- Let’s create the truth table:

| A | B | A ∧ B | B ∧ A |
|-------|-------|-------|-------|
| True | True | True | True |
| True | False | False | False |
| False | True | False | False |
| False | False | False | False |

- From the table, we see that A ∧ B = B ∧ A for all combinations of A and B. Thus, the AND operator satisfies the commutative law.

4. **Truth Table for OR Operator**:
- The OR operator (denoted as ∨) is true if at least one operand is true.
- Let’s create the truth table:

| A | B | A ∨ B | B ∨ A |
|-------|-------|-------|-------|
| True | True | True | True |
| True | False | True | True |
| False | True | True | True |
| False | False | False | False |

- From the table, we see that A ∨ B = B ∨ A for all combinations of A and B. Thus, the OR operator also satisfies the commutative law.

5. **Truth Table for Biconditional Operator**:
- The Biconditional operator (denoted as ↔) is true if both operands are the same.
- Let’s create the truth table:

| A | B | A ↔ B | B ↔ A |
|-------|-------|-------|-------|
| True | True | True | True |
| True | False | False | False |
| False | True | False | False |
| False | False | True | True |

- From the table, we see that A ↔ B = B ↔ A for all combinations of A and B. Thus, the Biconditional operator also satisfies the commutative law.

6. **Conclusion**:
Since all three operators (AND, OR, and Biconditional) satisfy the commutative law, the answer to the question is that all of the above connectives satisfy the commutative law.

### Final Answer:
All of the above connectives satisfy the commutative law.