We come now to the special cases of dyadic relations known as functions. It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory.
To begin, let us recall the definition of a local flag 
In the case of a dyadic relation 
![\begin{array}{lllll} u \star L & = & L_{u @ X} & = & L_{u @ 1} \\[6pt] L \star v & = & L_{v @ Y} & = & L_{v @ 2} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blllll%7D++u+%5Cstar+L+%26+%3D+%26+L_%7Bu+%40+X%7D+%26+%3D+%26+L_%7Bu+%40+1%7D++%5C%5C%5B6pt%5D++L+%5Cstar+v+%26+%3D+%26+L_%7Bv+%40+Y%7D+%26+%3D+%26+L_%7Bv+%40+2%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{lllll} u \star L & = & L_{u @ X} & = & L_{u @ 1} \\[6pt] L \star v & = & L_{v @ Y} & = & L_{v @ 2} \end{array}")
In light of these conventions, the local flags of a dyadic relation 
![\begin{array}{lll} u \star L & = & L_{u @ X} \\[6pt] & = & \{ (u, y) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ u \in X. \\[9pt] L \star v & = & L_{v @ Y} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++u+%5Cstar+L+%26+%3D+%26+L_%7Bu+%40+X%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+%28u%2C+y%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bthe+ordered+pairs+in%7D%7E+L+%7E%5Ctext%7Bthat+are+incident+with%7D%7E+u+%5Cin+X.++%5C%5C%5B9pt%5D++L+%5Cstar+v+%26+%3D+%26+L_%7Bv+%40+Y%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+%28x%2C+v%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bthe+ordered+pairs+in%7D%7E+L+%7E%5Ctext%7Bthat+are+incident+with%7D%7E+v+%5Cin+Y.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{lll} u \star L & = & L_{u @ X} \\[6pt] & = & \{ (u, y) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ u \in X. \\[9pt] L \star v & = & L_{v @ Y} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y. \end{array}")
The following definitions are also useful:
![\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ y \in Y : (u, y) \in L \} \\[6pt] & = & \text{the elements of}~ Y ~\text{that are}~ L\text{-related to}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++u+%5Ccdot+L+%26+%3D+%26+%5Cmathrm%7Bproj%7D_2+%28u+%5Cstar+L%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+y+%5Cin+Y+%3A+%28u%2C+y%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bthe+elements+of%7D%7E+Y+%7E%5Ctext%7Bthat+are%7D%7E+L%5Ctext%7B-related+to%7D%7E+u.++%5C%5C%5B9pt%5D++L+%5Ccdot+v+%26+%3D+%26+%5Cmathrm%7Bproj%7D_1+%28L+%5Cstar+v%29++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+x+%5Cin+X+%3A+%28x%2C+v%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bthe+elements+of%7D%7E+X+%7E%5Ctext%7Bthat+are%7D%7E+L%5Ctext%7B-related+to%7D%7E+v.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1 "\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ y \in Y : (u, y) \in L \} \\[6pt] & = & \text{the elements of}~ Y ~\text{that are}~ L\text{-related to}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v. \end{array}")
A sufficient illustration is supplied by the earlier example
 |
(35) |
The local flag 
 |
(36) |
The local flag 
 |
(37) |
Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry
Pingback: Peirce’s 1870 “Logic Of Relatives” • Overview | Inquiry Into Inquiry
Pingback: Peirce’s 1870 “Logic Of Relatives” • Comment 1 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 4 | Inquiry Into Inquiry