Checking Binary Relations by Definition

What is Binary Relation

Binary relation over sets X and Y is a new set of ordered pairs (x, y) consisting of elements x in X and y in Y.

Let $\bbox[white]{(x, y) \in \tau \equiv x \tau y}$

Сhecking of belonging to:

Reflexive relation: $\bbox[white]{\forall x \in M: x \tau x}$
Symmetric relation: $\bbox[white]{\forall x, y \in M: x \tau y \Rightarrow y \tau x}$
Antisymmetric relation: $\bbox[white]{\forall x, y \in M: x \tau y, y \tau x \Rightarrow x = y}$
Transitive relation: $\bbox[white]{\forall x, y, z \in M: x \tau y, y \tau z \Rightarrow x \tau z}$
Equivalence relation: if Binary Relation is Reflexive, Symmetric and Transitive
Order relation: if Binary Relation is Reflexive, Antisymmetric and Transitive

Example

If $\bbox[white]{M = \{ 1, 2, 3, 4 \}}$ and $\bbox[white]{x \tau y \Leftrightarrow \text{(x + y is even sum)}}$

----------M-------------------------------------------------------------------------------
(1, 2, 3, 4)

----------M * M---------------------------------------------------------------------------
(1,1)   (1,2)   (1,3)   (1,4)
(2,1)   (2,2)   (2,3)   (2,4)
(3,1)   (3,2)   (3,3)   (3,4)
(4,1)   (4,2)   (4,3)   (4,4)

----------M * M (with tau)----------------------------------------------------------------
(1,1)   -       (1,3)   -
-       (2,2)   -       (2,4)
(3,1)   -       (3,3)   -
-       (4,2)   -       (4,4)

----------properties of binary relations--------------------------------------------------
reflexive_relation      +
symmetric_relation      +
antisymmetric_relation  -       (4,2) in taus, and (2,4) in taus, but 4 != 2
transitive_relation     +

----------types of binary relations-------------------------------------------------------
equivalence relation    +
order relation          -

llirrikk / binary-relations

Checking Binary Relations by Definition

What is Binary Relation

Example

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