Relations and Functions
Relations and Functions
Recap from Last Time
Reflexivity
Some relations always hold from any element to itself.
Examples:
x = x for any x. A A for any set A. x x for any x.
Relations of this sort are called reflexive. Formally speaking, a binary relation R over a set A
is reflexive if the following is true: a A. aRa
("Every element is related to itself.")
Reflexivity Visualized
a A. aRa ("Every element is related to itself.")
Symmetry
In some relations, the relative order of the objects doesn't matter.
Examples:
If x = y, then y = x. If x y, then y x.
These relations are called symmetric. Formally: a binary relation R over a set A is called
symmetric if a A. b A. (aRb bRa)
("If a is related to b, then b is related to a.")
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