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Let $Y \subseteq \mathbb{R}^n$. I have the following statement: "$Y$ satisfies constant returns to scale if $y \in Y$ implies $\forall \alpha \ge 0$ it follows that $\alpha y \in Y$."

I am trying to find the definition of $Y$ does not satisfy constant returns to scale, so I am trying to negate the statement. What is the correct negation? Is it: "$Y$ does not satisfy constant returns to scale if there exists a $y \in Y$ and there exists a $\alpha \ge 0$ such that $\alpha y \not\in Y$"?

If my negation is correct, can someone show me how to arrive at it using propositional logic? (i.e., using quantifiers etc)

elbarto
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1 Answers1

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It is correct. Using quantifiers, the original statement is $$ \forall y \in Y, \forall \alpha\in [0,\infty), \alpha y \in Y $$ The negation is $$ \exists y \in Y, \exists \alpha\in [0,\infty), \alpha y \not\in Y $$

Gribouillis
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