Axiom of Extensionality, I understand that if two sets have exactly the same members they are equal. However why it is called Extensionality? Why not equality?
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I also wonder about this, since it seems to me that the axiom of extensionality is there to define "what equality is for sets", instead of what a set is. – soomakan. Nov 11 '21 at 17:41
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"Extension" refers to a set being defined by its content, as opposed to "intension" which is a term to say it is defined by some form of specification.
Let $A = \{x \text{ in } \mathbb{R} \text{ such that} -1 \leq x \leq +1\}$
Let $B = \{y \text{ in } \mathbb{R} \text{ such that } y = \sin(x) \text{ for some } x \text{ in } \mathbb{R}\}$.
The sets have different intensions (specifications), but contain the same elements and therefore have the same extensions. So (according to the "Axiom of Extension") $A = B$.
Christian E. Ramirez
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Tom Collinge
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OK. My understanding as a layman.
Element x is present in A. Element X is present in B. "Extension means" the very presence of x in another set B, Not just only in A. – Vinodh Feb 11 '14 at 13:50I understand the specification: The set of students in Music Class "can be" the same for another Drawing class. However the purpose of the class is Drawing, rather than music. Thanks a lot! -
A (much) fuller explanation of extension and intension can be found at http://plato.stanford.edu/entries/logic-intensional/ – Tom Collinge Feb 11 '14 at 14:08
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Quoting from the SEP link above "The two phrases, “morning star” and “evening star” may designate the same object, but they do not have the same meaning. Meanings, in this sense, are often called intensions, and things designated, extensions.". Thanks for the link. – Vinodh Feb 11 '14 at 16:45
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Because it says a set is determined by its extension.
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Thanks. When we say, set A = set B, what is extension here? The elements of A belonging to another set as well is what we mean by extension? – Vinodh Feb 09 '14 at 13:58