Existence Theorem

Pronunciation: /ɪgˈzɪs təns ˈθiə rəm/ ?

An existence theorem is a theorem that proves the existence of an entity or entities without telling how many entities there are or how to find them.[1] On example of an existence theorem is that for all polynomials, if a value of the polynomial is positive for one value of x, and negative for another value of x, then the value of the polynomial must be zero somewhere in between the two values of x.

In figure 1, the points (-2.5,0.875) and (-1,-4) are plotted. Since f(-2.5) is positive, and f(-1) is negative, then for some value of x, -2.5 < x < -1, f(x)=0. Notice that this theorem does not tell is for how many values of x that f(x) is zero, nor how to find the value of f(x).

Graph of x^3+4x^2+x-6 with the points (-2,0.875) and (-1,4) plotted.
Figure 1

References

  1. existence theorem. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2009-03-12). http://www.merriam-webster.com/dictionary/existence theorem.
  2. Cupillari, Antonella. Nuts and Bolts of Proof: An Introduction to Mathematical Proofs, 3rd edition, pp 58-60. Academic Press. (Accessed: 2010-01-11).
  3. Daniel J. Velleman. How to Prove It: A Structured Approach, 2nd edition, pp 146-153. Cambridge University Press, January 16, 2006. (Accessed: 2010-02-02).

Printed Resources

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Existence Theorem. 2010-02-02. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/e/existencetheorem.html.

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2010-02-02: Added "References" (McAdams, David.)
2009-01-04: Expanded 'More Information' (McAdams, David.)
2008-06-10: Initial version (McAdams, David.)
2008-06-07: Corrected spelling (McAdams, David.)

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