When performing these arithmetic operations with five-bit signed representations, it's important to remember that:

• In five-bit signed numbers, the first bit is the sign bit:
  • 0 = positive
  • 1 = negative
• The range of values for five-bit signed numbers is $-16$ to $+15$.
  • Positive numbers range from $00000$ (0) to $01111$ (+15)
  • Negative numbers range from $11111$ (-1) to $10000$ (-16)

We'll convert each binary number to its decimal equivalent, perform the operations, and check for overflow. Overflow happens when the result is outside the range $[-16, 15]$.

a) $10110 + 01101$

1. Convert to decimal:

   • $10110 = -10$ (two's complement)
   • $01101 = +13$

2. Perform the addition: $-10 + 13 = 3$

3. Check for overflow:

   • The result, $3$, is within the valid range $[-16, 15]$, so no overflow.

Result: $3$, no overflow.

b) $11001 + 00101$

1. Convert to decimal:

   • $11001 = -7$ (two's complement)
   • $00101 = +5$

2. Perform the addition: $-7 + 5 = -2$

3. Check for overflow:

   • The result, $-2$, is within the valid range $[-16, 15]$, so no overflow.

Result: $-2$, no overflow.

c) $10110 - 01101$

For subtraction, convert the subtrahend ($01101$) to its two's complement form and add it.

1. Convert to decimal:

   • $10110 = -10$ (two's complement)
   • $01101 = +13$

2. Find two's complement of $01101$:

   • Flip the bits: $10010$
   • Add 1: $10011 = -13$ (two's complement)

3. Perform the addition: $-10 + (-13) = -23$

4. Check for overflow:

   • The result, $-23$, is outside the valid range $[-16, 15]$, so overflow occurs.

Result: Overflow occurs because $-23$ is out of range.

d) $11111 - 01011$

For subtraction, convert the subtrahend ($01011$) to its two's complement form and add it.

1. Convert to decimal:

   • $11111 = -1$ (two's complement)
   • $01011 = +11$

2. Find two's complement of $01011$:

   • Flip the bits: $10100$
   • Add 1: $10101 = -11$ (two's complement)

3. Perform the addition: $-1 + (-11) = -12$

4. Check for overflow:

   • The result, $-12$, is within the valid range $[-16, 15]$, so no overflow.

Result: $-12$, no overflow.

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Would you like any details or further clarifications? Here are some related questions for further exploration:

1. What are the benefits of using two's complement for signed number representation?
2. How do you detect overflow in signed binary addition?
3. What would happen if we used 6-bit signed numbers instead of 5-bit?
4. Can you explain the difference between one's complement and two's complement representation?
5. How is two's complement subtraction different from addition?

Tip: In two's complement, to negate a binary number, you simply flip the bits and add 1, making arithmetic operations like subtraction simpler!