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In [[axiomatic set theory]] and the branches of [[logic]], [[mathematics]], and [[computer science]] that use it, the '''axiom of extensionality''', or '''axiom of extension''', is one of the [[axiom]]s of [[Zermelo–Fraenkel set theory]].
 
== Formal statement ==
In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:
 
:<math>\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)</math>
 
or in words:
:[[Given any]] [[Set (mathematics)|set]] ''A'' and any set ''B'', if for every set ''X'', ''X'' is a member of ''A'' [[if and only if]] ''X'' is a member of ''B'', then ''A'' is [[equal (math)|equal]] to ''B''.
:(It is not really essential that ''X'' here be a ''set''&nbsp;&mdash; but in '''ZF''', everything is. See '''[[#In set theory with ur-elements|Ur-elements]]''' below for when this is violated.)
 
The converse, <math>\forall A \, \forall B \, (A = B \Rightarrow \forall X \, (X \in A \iff X \in B) )</math>, of this axiom follows from the substitution property of [[equality (mathematics)|equality]].
 
== Interpretation ==
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that ''A'' and ''B'' have precisely the same members.
Thus, what the axiom is really saying is that two sets are equal [[if and only if]] they have precisely the same members.
The essence of this is:
:A set is determined uniquely by its members.
 
The axiom of extensionality can be used with any statement of the form
<math>\exist A \, \forall X \, (X \in A \iff P(X) \, )</math>,
where ''P'' is any unary [[Predicate (mathematics)|predicate]] that does not mention ''A'', to define a unique set <math>A</math> whose members are precisely the sets satisfying the predicate <math>P</math>.
We can then introduce a new symbol for <math>A</math>; it's in this way that [[definition]]s in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.
 
The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.
However, it may require modifications for some purposes, as below.
 
== In predicate logic without equality ==
The axiom given above assumes that equality is a primitive symbol in [[predicate logic]].
Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a ''definition'' of equality.
Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is
:<math>\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow \forall Y \, (A \in Y \iff B \in Y) \, )</math>
and it becomes ''this'' axiom that is referred to as the axiom of extensionality in this context.
 
== In set theory with ur-elements ==
An [[ur-element]] is a member of a set that is not itself a set.
In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory.
Ur-elements can be treated as a different [[logical type]] from sets; in this case, <math>B \in A</math> makes no sense if <math>A</math> is an ur-element, so the axiom of extensionality simply applies only to sets.
 
Alternatively, in untyped logic, we can require <math>B \in A</math>  to be false whenever <math>A</math> is an ur-element.
In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the [[empty set]].
To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:
 
:<math>\forall A \, \forall B \, ( \exist X \, (X \in A) \Rightarrow [ \forall Y \, (Y \in A \iff Y \in B) \Rightarrow A = B ] \, ).</math>
 
That is:
:Given any set ''A'' and any set ''B'', ''if ''A'' is a nonempty set'' (that is, if there exists a member ''X'' of ''A''), ''then'' if ''A'' and ''B'' have precisely the same members, then they are equal.
 
Yet another alternative in untyped logic is to define <math>A</math> itself to be the only element of <math>A</math>
whenever <math>A</math> is an ur-element.  While this approach can serve to preserve the axiom of extensionality, the [[axiom of regularity]] will need an adjustment instead.
 
== See also ==
* [[Extensionality]] for a general overview.
 
== References ==
*[[Paul Halmos]], ''Naive set theory''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).
*[[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''.  Springer.  ISBN 3-540-44085-2.
*[[Kenneth Kunen|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier.  ISBN 0-444-86839-9.
 
{{Set theory}}
 
[[Category:Axioms of set theory]]
[[Category:Urelements]]

Revision as of 21:18, 25 February 2014

Each day application of a moisturizer will keep psoriasis at bay. Non-Steroid Topical Medication- Various other ointments may be applied topically, which do not include steroids as an active ingredient. Mahonia aquifolium offers real hope to thousands of Psoriasis sufferers and is, without doubt, the safest & most effective herbal psoriasis treatment available. Keep way from eggs and animal fats as much as you can. An autoimmune disease, it is frequently hereditary, and has been linked to obesity, heart disease and diabetes.

What is considered normal changes as a person ages. My family laugh when they see how excited I get about having knuckles now instead of great patches of scales. After a spray tan, you must not shower for a minimum of four hours; this allows the DHA to take full effect. This ingredient is a specialized honey that comes from New Zealand also. Another frequent scalp condition is seborrheic dermatitis.

Psoriasis commonly affects the skin in the elbows, knees, and scalp. I began to feel self-conscious about my appearance and had a poor self-image that stemmed from fear of public rejection. They are the best moisturizers but are less pleasant to use because they are greasy and quite sticky. Most people who battle with a psoriasis problem also have skin which is extremely dry and flaky. DNA testing for genetic health risks can be now used to determine your risk for any genetic disease.

Hyper-pigmentation can be natural as in a birthmark that is a darker color of skin than your normal skin color. A physical therapist may also help in the work outs. Truth be told, it is calming to the point that specialists have really observed that it "switches off" the anxiety genes. It has also been observed that most of the people suffering from Psoriasis suffer from sexual recession because of the painful realization of their physical symptoms. is also used in more than 100 cosmetic formulations that appear in a wide range of products, including hair tonics, anti-dandruff products, and perfumes.

The major symptoms are reddening, thickening of the skin patches, itching, burning and scales formation. And while doctors will prescribe different medications and creams to help men and women deal with this issue most doctors will tell you that there's no cure for this condition. Applying these creams is most effective when used during nighttime, so the body can absorb it more easily, and your feet are resting and does not experience any other harmful movements. These are very simple ingredients that make the inflammation and also the itchy skin absolutely typical and you'd by no means really feel discomfort or under no circumstances feel the urge to scratch your skin regularly. A few may appear to work mainly because psoriasis is characterized by flare-ups and remissions.

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