The power set of a set forms an abelian group when considered with the operation of symmetric difference ( with the empty set as the identity element and each set being its own inverse ) and a commutative monoid when considered with the operation of intersection.
32.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed.
33.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed.
34.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed.
35.
The symmetric difference quotient is employed as the method of approximating the derivative in a number of calculators, including TI-82, TI-83, TI-84, TI-85 all of which use this method with " h " = 0.001.
36.
It is already guaranteed : After doing the symmetric difference for a path, none of its vertices could be considered again just because the Dist [ Pair _ V [ v ] ] will not be equal to Dist [ u ] + 1 ( It would be exactly Dist [ u ] ).
37.
Both approaches use the observation that in claw-free graphs, no vertex can have more than two neighbors in an independent set, and so the symmetric difference of two independent sets must induce a subgraph of degree at most two; that is, it is a union of paths and cycles.
38.
For p = 2, the symmetric difference of two distinct index 2 subgroups ( which are necessarily normal ) gives the third point on the projective line containing these subgroups, and a group must contain 0, 1, 3, 7, 15, \ ldots index 2 subgroups it cannot contain exactly 2 or 4 index 2 subgroups, for instance.
39.
This implies triangle inequality : the symmetric difference of " A " and " C " is contained in the union of the symmetric difference of " A " and " B " and that of " B " and " C " . ( But note that for the diameter of the symmetric difference the triangle inequality does not hold .)
40.
This implies triangle inequality : the symmetric difference of " A " and " C " is contained in the union of the symmetric difference of " A " and " B " and that of " B " and " C " . ( But note that for the diameter of the symmetric difference the triangle inequality does not hold .)
