1. **Problem:** Prove that for any natural number $k$, $1 \cdot k = k$. 2. **Formula and rules:** Multiplication by 1 is the identity operation in natural numbers, meaning multiplying any number by 1 leaves it unchanged. 3. **Proof:** - By definition of multiplication, $1 \cdot k$ means adding $k$ to itself 1 time. - So, $1 \cdot k = k$. 4. **Explanation:** Multiplying by 1 does not change the value of $k$ because 1 is the multiplicative identity. 1. **Problem:** Prove that for any natural numbers $k$ and $n$, $k + n = n + k$. 2. **Formula and rules:** This is the commutative property of addition, which states that the order of addition does not affect the sum. 3. **Proof:** - By the commutative property, $k + n = n + k$. - This can be shown by induction or by the definition of addition in natural numbers. 4. **Explanation:** The sum remains the same regardless of the order of the addends. 1. **Problem:** Show that $k + 1 = k^+$ (the successor of $k$). 2. **Formula and rules:** The successor function $k^+$ is defined as the next natural number after $k$, which is $k + 1$. 3. **Proof:** - By definition, $k^+ = k + 1$. 4. **Explanation:** Adding 1 to $k$ gives the next natural number, which is the successor of $k$. **Final answers:** - $1 \cdot k = k$ - $k + n = n + k$ - $k + 1 = k^+$