Test on Addition and subtraction in the binary system

Question 1: In the provided example of binary addition (1011₂ + 0101₂), the sum in the second column (2¹ position) after adding the carry-in and the original bits is 1, with no carry generated.

A. Yes

B. No

Explanation: According to the step-by-step explanation for the example, the sum in the second column (2¹ position) is '1 (carry) + 1 + 0 = 10₂'. This means the result for that column is 0, and a carry of 1 is generated to the next column, not a sum of 1 with no carry.

Question 2: According to the provided study materials, which of the following statements accurately describes a basic rule of binary addition?

A. 1 + 1 = 1 with a carry of 1.

B. 0 + 1 + 1 = 10₂ (0 with a carry of 1).

C. 1 + 1 + 1 = 10₂ (1 with a carry of 0).

D. 0 + 0 + 0 = 1.

Explanation: The study materials state that when adding two binary digits, '1 + 1 = 10₂ (which means 0 with a carry of 1 to the next column)'. Also, for three bits, '0 + 1 + 1 = 10₂ (0 with a carry of 1)' is a correct rule. The option '1 + 1 = 1 with a carry of 1' is incorrect because it should result in 0 with a carry of 1. The option '1 + 1 + 1 = 10₂ (1 with a carry of 0)' is incorrect, as it should be '1 + 1 + 1 = 11₂ (1 with a carry of 1)'. The option '0 + 0 + 0 = 1' is incorrect, as '0 + 0 + 0 = 0'.

Question 3: When subtracting binary numbers using two's complement, any carry-out from the most significant bit is discarded.

A. Yes

B. No

Explanation: According to the study materials, in step 4 of 'Subtraction using Two's Complement': 'If there is a carry-out from the most significant bit (the n-th bit), discard it. The remaining n bits are the result.'

Question 4: When subtracting 0101₂ from 1011₂ using 4-bit two's complement, the result 0110₂ is obtained after discarding the carry-out from the most significant bit.

A. Yes

B. No

Explanation: To subtract 0101₂ from 1011₂: First, find the two's complement of 0101₂. The one's complement of 0101₂ is 1010₂, and adding 1 gives 1011₂. Then, add 1011₂ (which is -0101₂) to 1011₂: 1011₂ + 1011₂ = (1)0110₂. Discarding the carry-out (1) from the leftmost bit leaves 0110₂ as the result.

Question 5: According to the study materials, which of the following statements accurately describes the overflow condition when subtracting 7 - (-2) using 4-bit two's complement arithmetic?

A. The subtraction yields 1001₂, which is decimal -7, indicating an overflow because the expected result of 9 is outside the -8 to +7 range.

B. The carry-in to the MSB was 0, and the carry-out was 1, which are different, thus indicating an overflow.

C. The operation 7 - (-2) subtracts a negative number from a positive number, and the result is negative, which signals an overflow.

D. The result 1001₂ converts to decimal 9, which is within the representable range for 4-bit two's complement.

Explanation: The study materials state that for the calculation 7 - (-2) using 4-bit signed numbers, the sum 0111₂ + 0010₂ equals 1001₂. This result (1001₂) is decimal -7, but 7 - (-2) should be 9. Since 9 is outside the range of 4-bit signed numbers (-8 to +7), an overflow has occurred (Option 0). The materials also explicitly mention that the carry-in to the MSB was 0, and the carry-out was 1, and since these two carries are different, an overflow is indicated (Option 1). Furthermore, the study materials define overflow as occurring 'if you subtract a negative number from a positive number and the result is negative', which is precisely what happens when 7 (positive) - (-2) (negative) results in -7 (negative) (Option 2). Option 3 is incorrect because the result 1001₂ is decimal -7, not 9, and 9 is outside the -8 to +7 range.