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In learning math, we always encounter definitions. My question is whether the inverse proposition of a definition is always true.

For example, we have the definition of a symmetric matrix, which goes 'If $A$=$A^T$, then $A$ is symmertic.' It is obvious that when we have a symmetric matrix, it always satisfies $A$=$A^T$.

But if we use logic to explain, I have a problem. The definition part can be written as 'If $p$, then $q$', whose truth can not indicate the inverse 'if $q$, then $p$' is also true.

Andrew_Ren
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    The "if" in the context of a definition actually means "if and only if". The definition is stating the meaning of the new terminology, not just giving a logical proposition (even though that's what it looks like if you take it literally). – Karl Jul 20 '22 at 15:25
  • Also, by "inverse" you mean converse. – Karl Jul 20 '22 at 15:26
  • Thank you. It helps a lot! – Andrew_Ren Jul 20 '22 at 15:34
  • Definitions are always "if and only if". This is a semantic detail that originates, probably, in the spoken language. For example, say you wish to define the term "even". You'd say, "a number is called even if it can be divided by two with no remainder". – Roy Sht Jul 20 '22 at 15:34
  • I don't think the definition is actually written as you describe: typically, it goes "A is symmetric if $A=A^T$" rather than ""if A is symmetric, then $A=A^T$" or "if $A=A^T$, then A is symmetric" (that is, the word "then" is generally not used). But yes, in definitions, "if" means "iff": check out the screencap here: Why every definition is an "iff"-type statement?. – ryang Jul 20 '22 at 16:13
  • Its actually always if and only if – Wuestenfux Jul 20 '22 at 16:58

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