Proving the Equation a ⊕ b a b - 2a ∧ b Using Binary Arithmetic

In the realm of binary arithmetic, the relationship between the bitwise operations XOR and AND can be expressed elegantly through the equation:

Definitions

XOR#124;⊕: The bitwise XOR operation results in a binary value where each bit is set to 1 if the corresponding bits of the operands are different and 0 if they are the same. AND#124;("land": The bitwise AND operation results in a binary value where each bit is set to 1 only if both corresponding bits of the operands are 1.

Proof Steps

Let's break down the equation b CrossProduct a ⊕ b - 2a ∧ b using the fundamental definitions and properties of binary arithmetic.

Understanding Binary Addition

When adding two binary numbers a and b, it's important to consider two components:

The sum of the bits (without considering carries). The carry that results from the addition of bits.

Decomposing the Addition

The sum b can be expressed in terms of the XOR and AND operations:

The XOR operation a ⊕ b gives us the sum of the bits without considering the carry. The AND operation a ∧ b gives us the bits that will produce a carry.

Formulating the Carry

When two bits are added, a carry is generated when both bits are 1. The AND operation a ∧ b captures this scenario. Each carry from the addition effectively increases the sum by 2 for every bit position where the carry occurs. Thus, to account for the carries, we multiply the result of the AND operation by 2.

Combining the Results

Therefore, the total sum can be expressed as:

a b a ⊕ b - 2a ∧ b

Verification

To verify this, let's examine a few examples:

Example 1

Let a 3 (binary: 112) and b 5 (binary: 1012).

a b 3 5 8 a ⊕ b 3 ⊕ 5 6 (binary: 1102) a ∧ b 3 ∧ 5 1 (binary: 0012)

Therefore, 2a ∧ b 2 × 1 2.

Plugging into the equation:

a ⊕ b - 2a ∧ b 6 - 2 8

This confirms the equation holds: 8 8.

Example 2

Let a 2 (binary: 102) and b 3 (binary: 112).

a b 2 3 5 a ⊕ b 2 ⊕ 3 1 (binary: 012) a ∧ b 2 ∧ 3 2 (binary: 102)

Therefore, 2a ∧ b 2 × 2 4.

Plugging into the equation:

a ⊕ b - 2a ∧ b 1 - 4 5

This also confirms the equation holds: 5 5.

Conclusion

We have thus shown that b a ⊕ b - 2a ∧ b holds true through both conceptual understanding and verification with examples.