If all set members have a property, set has property {abstraction principle} {principle of abstraction}.
Elements added to sets {adjunction} can extend set properties.
Sets have a highest member {upper bound, set} {bound}. Sets have a lowest member {lower bound, set}. Sets can have minimum at highest member {greatest lower bound, set}. Sets can have maximum at lowest member {least upper bound, set}.
Ordered-set elements can advance one place {cyclic permutation}. Cyclic permutations can have only two elements {transposition, set}.
Two sets are equal if and only if sets have same members {principle of extension} {extension principle}.
The identity relation can apply to domain subset {injection mapping}.
Sets can have two binary operations {integral domain}. Operations make commutative rings and have identity elements. First operation has identity element if second operation happens.
Open sets can contain no point {interior point} at boundary.
Two sets are equivalent if their elements pair {one-to-one correspondence}|.
Relations {ordered} between two or more things can have sequence. Sequence does not have to be complete {partial order}.
number
For two things, first {first coordinate} and second {second coordinate} make pair {ordered pair, set} or relation {binary relation}. First coordinate belongs to domain. Second coordinate belongs to range. Second coordinate derives from first coordinate {Cartesian product, pair}.
Relations can be among many things {ordered n-tuple} {n-ary relation}, with many coordinates.
types
Order can be linear {simple order} {linear order}. All set members can have simple order {chain, set}. Non-empty sets can have unique lowest members {well-ordered set}.
value
Sets can have unique lowest {least element} member. Sets can have unique highest {greatest element} member. Maximum at lowest member {least upper bound, element} and minimum at highest member {greatest lower bound, element} bound interval.
Sets can split {partition, set}| into mutually exclusive subsets.
Relation between second coordinate and first coordinate can be image of relation between first coordinate and second coordinate {symmetric}. Relations can be reflexive and transitive {antisymmetric}, so both members are equal.
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Date Modified: 2022.0225