The correct answer for building a half adder using the minimum number of gates is option 'C', which requires an EX-OR gate and an AND gate. Let's understand why this is the optimal solution.

1. Half Adder:
A half adder is a digital circuit that performs addition of two single-bit numbers. It has two inputs, A and B, and two outputs, Sum (S) and Carry (C). The truth table for a half adder is as follows:

| A | B | S | C |
|-----|-----|-----|-----|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |

2. EX-OR Gate:
An EX-OR gate, also known as an exclusive OR gate, gives a high output (1) when the number of high inputs is odd. It gives a low output (0) when the number of high inputs is even. The truth table for an EX-OR gate is as follows:

| A | B | Output |
|-----|-----|---------|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |

3. AND Gate:
An AND gate gives a high output (1) only when all the inputs are high. The truth table for an AND gate is as follows:

| A | B | Output |
|-----|-----|---------|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |

4. Building a Half Adder using an EX-OR Gate and an AND Gate:
To build a half adder, we need to calculate the Sum (S) and the Carry (C) based on the inputs A and B.

- The Sum (S) can be obtained by connecting the inputs A and B to an EX-OR gate.
- The Carry (C) can be obtained by connecting the inputs A and B to an AND gate.

By connecting the outputs of the EX-OR gate and the AND gate, we can obtain the required outputs for a half adder. This combination provides the correct values for S and C according to the truth table of a half adder.

Hence, the correct answer is option 'C', which requires an EX-OR gate and an AND gate to build a half adder.