Sign function


THEOREM    \(\forall x \in \mathbb{R} : \sgn(\sgn(x)) = \sgn(x)\)

Proof available

THEOREM    \(\max(\sgn(x), \sgn(y)) = \sgn(\max(x, y))\)

Proof available

THEOREM    \(\min(\sgn(x), \sgn(y)) = \sgn(\min(x, y))\)

Proof available

THEOREM    \(\forall a,b \in \mathbb{R} : \sgn(ab) = \sgn(a)\sgn(b)\)
An Introduction to Proofs and the Mathematical Vernacular
Problem 1.5a (page 5)

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THEOREM    \(\forall a \in \mathbb{R} : a \ne 0 \Rightarrow \sgn(a) = \sgn\left(\dfrac{1}{a}\right)\)
An Introduction to Proofs and the Mathematical Vernacular
Problem 1.5b (page 5)

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THEOREM    \(\forall a \in \mathbb{R} : \sgn(-a) = -\sgn(a)\)
An Introduction to Proofs and the Mathematical Vernacular
Problem 1.5c (page 5)

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THEOREM    \(\forall a \in \mathbb{R} : a \ne 0 \Rightarrow \sgn(a) = \dfrac{a}{|a|}\)
An Introduction to Proofs and the Mathematical Vernacular
Problem 1.5d (page 5)

Proof unavailable

THEOREM   

Proof unavailable