1. **Problem Statement:** We will answer each sub-question related to number systems, binary operations, digital logic, and Boolean algebra. --- 1.1 **Symbols of the octal number system:** The octal system uses digits from 0 to 7. **Answer:** $\{0,1,2,3,4,5,6,7\}$ --- 1.2 **Decimal value of a binary 1 in the second position to the right of the radix comma:** The radix comma separates integer and fractional parts. The first position right of the radix is $2^{-1} = \frac{1}{2}$. The second position right of the radix is $2^{-2} = \frac{1}{4} = 0.25$. **Answer:** $0.25$ --- 1.3 **Biggest symbol in the hexadecimal number system:** Hexadecimal digits are $0-9$ and $A-F$ where $A=10$ and $F=15$. The biggest symbol is $F$. **Answer:** $F$ --- 1.4 **Two's complement of $10110101_2$:** Step 1: Invert all bits: $10110101 \to 01001010$ Step 2: Add 1: $$01001010 + 1 = 01001011$$ **Answer:** $01001011_2$ --- 1.5 **Which binary counter uses output from previous flip-flop to clock the next?** - Synchronous counters use a common clock. - Asynchronous counters use output from previous flip-flop as clock for next. **Answer:** Asynchronous binary counter --- 1.6 **Number of select lines for a 4:1 multiplexer:** Number of select lines $= \log_2(4) = 2$ **Answer:** 2 --- 1.7 **Main difference between full adder and half adder:** - Half adder adds two bits, outputs sum and carry. - Full adder adds three bits (including carry-in), outputs sum and carry-out. **Answer:** Full adder handles carry-in; half adder does not. --- 1.8 **Small scale integration (SSI):** SSI refers to integrated circuits with a small number of gates (typically less than 100). Used for simple logic functions. **Answer:** Integration of a few logic gates on a single chip. --- 1.9 **Simplify Boolean expressions:** 1.9.1 $(A + \overline{A}) + \overline{B}B$ - $A + \overline{A} = 1$ - $\overline{B}B = 0$ - So expression $= 1 + 0 = 1$ 1.9.2 $AC̅ + 1 + A$ - Anything OR 1 is 1 - So expression $= 1$ 1.9.3 $BCD + B + BC̅D̅$ - $B + BCD = B$ - $B + BC̅D̅ = B$ - So expression $= B$ 1.9.4 $AB + AB̅\overline{A}$ - Note $\overline{A}A = 0$ - So $AB + AB̅\overline{A} = AB + 0 = AB$ --- 1.10 **Simplify Boolean expressions from Karnaugh maps:** 1.10.1 Map values: - Rows CD: 00,01,11,10 - Columns AB: 00,01,11,10 - Values: - Row 00: 1 1 1 1 - Row 01: 1 0 0 1 - Row 11: 1 0 0 1 - Row 10: 1 1 1 1 **Grouping:** - Entire first and last rows are 1s (rows 00 and 10) - First and last columns are 1s in all rows - Group 1: All 1s in rows 00 and 10 (covers all columns) - Group 2: First column (AB=00) all 1s - Group 3: Last column (AB=10) all 1s **Simplified expression:** - Group 1: $\overline{C}\overline{D} + C\overline{D} = \overline{D}$ - Group 2: $\overline{A}\overline{B}$ - Group 3: $A\overline{B}$ Combine: $$\overline{D} + \overline{A}\overline{B} + A\overline{B} = \overline{D} + \overline{B}(\overline{A} + A) = \overline{D} + \overline{B}$$ 1.10.2 Map values: - Row 00: 1 0 0 1 - Row 01: 1 0 0 1 - Row 11: 1 1 1 1 - Row 10: 1 0 0 1 **Grouping:** - Row 11 all 1s - First and last columns mostly 1s - Group 1: Row 11 (C D = 11) all columns - Group 2: First column (AB=00) all rows - Group 3: Last column (AB=10) all rows Simplified expression: - Row 11: $CD$ - First column: $\overline{A}\overline{B}$ - Last column: $A\overline{B}$ Combine: $$CD + \overline{B}(\overline{A} + A) = CD + \overline{B}$$ 1.10.3 Map values: - Row 00: 1 ∅ ∅ ∅ - Row 01: 1 0 1 1 - Row 11: ∅ 0 ∅ ∅ - Row 10: 1 1 1 1 **Grouping:** - Row 10 all 1s - Column 00 mostly 1s - Group 1: Row 10 (C D = 10) all columns - Group 2: Column 00 (AB=00) rows 00,01,10 Simplified expression: - Row 10: $C\overline{D}$ - Column 00: $\overline{A}\overline{B}$ Combine: $$C\overline{D} + \overline{A}\overline{B}$$ --- **Final answers:** 1.1: $\{0,1,2,3,4,5,6,7\}$ 1.2: $0.25$ 1.3: $F$ 1.4: $01001011_2$ 1.5: Asynchronous binary counter 1.6: 2 1.7: Full adder handles carry-in; half adder does not 1.8: Integration of a few logic gates on a single chip 1.9.1: 1 1.9.2: 1 1.9.3: $B$ 1.9.4: $AB$ 1.10.1: $\overline{D} + \overline{B}$ 1.10.2: $CD + \overline{B}$ 1.10.3: $C\overline{D} + \overline{A}\overline{B}$