Difference between revisions of "One-to-one correspondence"
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| − | A correspondence between elements of two sets in which: 1) to each element of the first set there corresponds a unique element of the second set; 2) to different elements of the first set there correspond different elements of the second set; and 3) each element of the second set has been put into correspondence with an element of the first set. The relation which holds between sets if and only if there is a one-to-one correspondence between them is symmetric (the mapping inverse to a one-to-one correspondence is a one-to-one correspondence) and is transitive (the product of one-to-one correspondences is a one-to-one correspondence). If each point | + | {{TEX|done}} |
| + | A correspondence between elements of two sets in which: 1) to each element of the first set there corresponds a unique element of the second set; 2) to different elements of the first set there correspond different elements of the second set; and 3) each element of the second set has been put into correspondence with an element of the first set. The relation which holds between sets if and only if there is a one-to-one correspondence between them is symmetric (the mapping inverse to a one-to-one correspondence is a one-to-one correspondence) and is transitive (the product of one-to-one correspondences is a one-to-one correspondence). If each point $x$ of an oriented straight line is put into correspondence with its distance from a given point $O$ (which is positive if the point is located in a positive direction from $O$, and is negative otherwise), the resulting correspondence is a one-to-one correspondence between the points on the straight line and the real numbers. | ||
Latest revision as of 15:45, 1 May 2014
A correspondence between elements of two sets in which: 1) to each element of the first set there corresponds a unique element of the second set; 2) to different elements of the first set there correspond different elements of the second set; and 3) each element of the second set has been put into correspondence with an element of the first set. The relation which holds between sets if and only if there is a one-to-one correspondence between them is symmetric (the mapping inverse to a one-to-one correspondence is a one-to-one correspondence) and is transitive (the product of one-to-one correspondences is a one-to-one correspondence). If each point $x$ of an oriented straight line is put into correspondence with its distance from a given point $O$ (which is positive if the point is located in a positive direction from $O$, and is negative otherwise), the resulting correspondence is a one-to-one correspondence between the points on the straight line and the real numbers.
Comments
See also Bijection.
References
| [a1] | H.B. Enderton, "Elements of set theory" , Acad. Press (1977) |
One-to-one correspondence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=One-to-one_correspondence&oldid=32076