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negation

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Could someone explain to me about what closed under addition and closed under scalar multiplication means? I have a patchy idea of what it is but how does it relates to A = {(x,y) | x^2 + y^2 <= 1}?

What does A stands for? What does the language implies?

Edit: My interpretation: Let's suppose there exists a field k with R^n where n = 2) and A is a subset of the field k.

An element is closed under addition iff an element, u

u^2+v^2 = <=1.

If u^2+v^2 <=1, then, u and v is a subset of A.

What does A stands for? What does the language implies?

Edit: My interpretation: Let's suppose there exists a field k with R^n where n = 2) and A is a subset of the field k.

An element is closed under addition iff an element, u

_{A}, and, v_{A}such thatu^2+v^2 = <=1.

If u^2+v^2 <=1, then, u and v is a subset of A.

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