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  1. #1
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    predicate logic

    “Exponentiation on reals has no left identity.” What does this mean exactly?
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  2. #2
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    Identity element

    Hello thehollow89
    Quote Originally Posted by thehollow89 View Post
    “Exponentiation on reals has no left identity.” What does this mean exactly?
    Exponentiation on reals takes a real number, $\displaystyle x$, and raises a base, $\displaystyle a$, say, to the power of $\displaystyle x$. So, using $\displaystyle \circ$ notation, exponentiation can be defined as the binary operation:

    $\displaystyle a\circ x = a^x$.

    A left identity is an element $\displaystyle i$ in the domain of
    $\displaystyle \circ$, such that $\displaystyle i \circ x = x, \forall x$ in the domain.

    So to say that exponentiation on reals has no left identity is to say that there is no real number $\displaystyle i$ for which $\displaystyle i^x = x, \forall x \in \mathbb{R}$.

    Grandad
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