Question

Prove that if f and g are functions then \( f \cap g \) is a function by showing that \( f \cap g = g|_A \) where A={x: g(x)=f(x)}

Answer 1

By that notation you mean the restriction of g to A, right? This is pretty simple. If \(g(x)\ne f(x)\) then the intersection is empty, and if g(x)=f(x) the intersection is g(x), so the intersection is exactly the restriction as written. Since the restriction of a function is itself a function, there you go.

Answer 2

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