To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:
Recalling a few definitions, the local flags of the relation
![\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}. \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++u+%5Cstar+L++%26+%3D+%26+L_%7Bu%5C%2C%40%5C%2C1%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5C%7B+%28u%2C+x%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bthe+ordered+pairs+in%7D%7E+L+%7E%5Ctext%7Bthat+have%7D%7E+u+%7E%5Ctext%7Bin+the+1st+place%7D.++%5C%5C%5B9pt%5D++L+%5Cstar+v++%26+%3D+%26+L_%7Bv%5C%2C%40%5C%2C2%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+have%7D%7E+v+%7E%5Ctext%7Bin+the+2nd+place%7D.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} u \star L & = & L_{u\,@\,1} \\[6pt] & = & \{ (u, x) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ u ~\text{in the 1st place}. \\[9pt] L \star v & = & L_{v\,@\,2} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that have}~ v ~\text{in the 2nd place}. \end{array}")
The applications of the relation
![\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ x \in X : (u, x) \in L \} \\[6pt] & = & \text{loved by}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{lover of}~ 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+x+%5Cin+X+%3A+%28u%2C+x%29+%5Cin+L+%5C%7D++%5C%5C%5B6pt%5D++%26+%3D+%26+%5Ctext%7Bloved+by%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%7Blover+of%7D%7E+v.++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002 "\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ x \in X : (u, x) \in L \} \\[6pt] & = & \text{loved by}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{lover of}~ v. \end{array}")
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