Definitions and iff

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Purpose of "definitions"

Suppose we make the following definition:

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

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

But also we use definitions as follows:

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

We see immediately:

• [ilmath]\forall X[D\iff P(X)][/ilmath]

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

Thus: [ilmath]X[/ilmath] is [ilmath]D[/ilmath] if and only if [ilmath]P(X)[/ilmath] holds

Examples

Let [ilmath]X[/ilmath] be (whatever), we say [ilmath]X[/ilmath] is [ilmath]D[/ilmath] if:

• [ilmath]P(X)[/ilmath] holds.

We get both:

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