Final answer:
Arithmetic operations on eight-bit 2's complement numbers follow standard arithmetic rules, with overflow determined by changes in the leading bit. For designing a circuit with NAND gates for a given function, boolean algebra and circuit design principles apply.
Step-by-step explanation:
When performing the operations involving eight-bit 2's complement numbers, you would apply the basic rules of arithmetic operations such as addition and subtraction while considering the sign of the numbers.
- Adding two positive numbers results in a positive number.
- Adding two negative numbers results in a negative number.
- When adding numbers of opposite signs, subtract the smaller absolute value from the larger absolute value, and the result takes the sign of the larger number.
- For subtraction, invert the sign of the number being subtracted and add following the above rules.
Arithmetic overflow occurs when the result of an operation exceeds the range representable by the given number format. In the case of adding two 2's complement numbers, if both operands are positive and the result has a leading 1, or both are negative and the result has a leading 0, then an overflow has occurred.
Concerning the circuit design question to implement the function f (x₁, x₂, x₃) = m(1, 3, 4, 6, 7) using NAND gates, you would need to determine the simplified Boolean expression using Karnaugh maps or other minimization techniques. Once you have the simplified expression, you can then design the circuit by appropriately connecting NAND gates to represent the expression.