View source for One-to-one correspondence

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068260/o0682601.png" /> of an oriented straight line is put into correspondence with its distance from a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068260/o0682602.png" /> (which is positive if the point is located in a positive direction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068260/o0682603.png" />, 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|Bijection]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.B. Enderton,   "Elements of set theory" , Acad. Press  (1977)</TD></TR></table>