Definitions and iff

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Purpose of "definitions"

Suppose we make the following definition:

• An $X$ is $D$ when it satisfies $P(X)$ (for some statement $P$), symbolically:
  • $\forall X[P(X)\implies D]$

We note this can be directly used to show $X$ is $D$ (we show $X$ satisfies $P$, or some property equivalent to or implying $P$, thus it is $D$)

But also we use definitions as follows:

• "Let $X$ be $D$" to mean $P(X)$ is true. Symbolically:
  • $\forall X[D\implies P(X)]$

We see immediately:

• $\forall X[D\iff P(X)]$

This makes perfect sense, as we'd want definitions to be equivalent to having some defined properties.

Thus: $X$ is $D$ if and only if $P(X)$ holds

Examples

Let $X$ be (whatever), we say $X$ is $D$ if:

• $P(X)$ holds.

We get both:

1. If we have a $Y$ for which $P(Y)$ holds $\implies$ $Y$ is a $D$
2. Let $Y$ be a $D$ $\implies$ $P(Y)$ holds