The problem consists of three sub-problems: (A) Complete a given logic circuit with appropriate gates and write the corresponding Boolean expression. The circuit controls a bicycle engine based on three inputs: A (helmet), B (key), and C (stand). (B) Complete a truth table for the given Boolean expression $F = (A+B)\overline{A}C$. (C) Convert the octal number $657_8$ to a hexadecimal number.

1. Problem Description

The problem consists of three sub-problems:

(A) Complete a given logic circuit with appropriate gates and write the corresponding Boolean expression. The circuit controls a bicycle engine based on three inputs: A (helmet), B (key), and C (stand).

(B) Complete a truth table for the given Boolean expression $F = (A+B)\overline{A}C$.

(C) Convert the octal number $657_8$ to a hexadecimal number.

2. Solution Steps

(A)

(i) Completing the logic circuit:

The problem states the bicycle engine only switches on when the rider wears a helmet (A), inserts the key (B), and releases the bicycle stand (C). This means A and B must both be true, and C must also be true simultaneously. Thus, $X$ should be an AND gate. Additionally, since releasing the bicycle stand means C should be 1 (true), we should actually take $\overline{C}$. Hence X should have an AND gate. And the final gate $Y$ will be a NAND gate, because we actually want $A \cdot B \cdot \overline{C}$. Thus, we need AND gate at $X$ and NOT gate before $C$ and AND gate at $Y$.

(ii) Boolean expression for the logic circuit:

From the completed circuit, the Boolean expression can be written as $F = (A \cdot B) \cdot \overline{C}$.

(B)

Completing the truth table for $F = (A+B)\overline{A}C$:

| A | B | C | A+B | $\overline{A}$ | $\overline{A}C$ | F = (A+B)$\overline{A}C$ |

|---|---|---|-----|-------|--------|-------------------|

| 0 | 0 | 0 | 0 | 1 | 0 | 0 |

| 0 | 0 | 1 | 0 | 1 | 1 | 0 |

| 0 | 1 | 0 | 1 | 1 | 0 | 0 |

| 0 | 1 | 1 | 1 | 1 | 1 | 1 |

| 1 | 0 | 0 | 1 | 0 | 0 | 0 |

| 1 | 0 | 1 | 1 | 0 | 0 | 0 |

| 1 | 1 | 0 | 1 | 0 | 0 | 0 |

| 1 | 1 | 1 | 1 | 0 | 0 | 0 |

(C)

Converting $657_8$ to hexadecimal:

Step 1: Convert the octal number to decimal.

$657_8 = (6 \times 8^2) + (5 \times 8^1) + (7 \times 8^0) = (6 \times 64) + (5 \times 8) + (7 \times 1) = 384 + 40 + 7 = 431_{10}$

Step 2: Convert the decimal number to hexadecimal.

Divide 431 by 16 repeatedly and record the remainders:

$431 \div 16 = 26$ remainder $15$ (F)

$26 \div 16 = 1$ remainder $10$ (A)

$1 \div 16 = 0$ remainder $1$ (1)

Read the remainders in reverse order: $1AF_{16}$

3. Final Answer

(A)

(i) Gate X: AND gate. Gate Y: AND gate combined with a NOT gate at input C (effectively a NAND functionality if viewed from A, B, and C directly).

(ii) $F = (A \cdot B) \cdot \overline{C}$

(B)

| A | B | C | F |

|---|---|---|---|

| 0 | 0 | 0 | 0 |

| 0 | 0 | 1 | 0 |

| 0 | 1 | 0 | 0 |

| 0 | 1 | 1 | 1 |

| 1 | 0 | 0 | 0 |

| 1 | 0 | 1 | 0 |

| 1 | 1 | 0 | 0 |

| 1 | 1 | 1 | 0 |

(C) $1AF_{16}$