The proposition "for all real numbers a and b, 2 · a · b = a² + b²" is false, as it is not valid for real numbers such that a ≠ b.
According to this question, we must infer on the nature of the sets related to the real numbers a and b contained in the algebraic expression 2 · a · b = a² + b². Then, we simplify the expression by algebraic handling:
2 · a · b = a² + b² Given
a² - 2 · a · b + b² = 0 Compatibility with addition / Existence of additive inverse / Modulative property
(a - b)² = 0 Perfect square binomial
± (a - b) = 0 Square root
± a = ± b Compatibility with addition / Existence of additive inverse / Modulative property
The algebraic expression is only valid when a = b, such that the equality is guaranteed. Hence, the expression is not valid for a ≠ b and the proposition is false.
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