Properties of Relations


Relations on a set

The relations in this section are defined on a set.

(The domain and the codomain are the same set.)


















Reflexive

What's a Reflexive relation?

	The relation contains (x,x) for every x in the domain.

	Every member of the domain is related to itself.

Determine whether each relation is Reflexive or not.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Reflexive or not.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















Symmetric

What's a Symmetric relation?

	If the relation contains (x,y),
	the relation must contain (y,x).

What's an Antisymmetric relation?

	If the relation contains (x,y),
	the relation must not contain (y,x),
	unless x and y are the same item.

Determine whether each relation is Symmetric, Antisymmetric, both, or neither.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Symmetric, Antisymmetric, both, or neither.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















Transitive

What's a Transitive relation?

	If the relation contains (x,y) and (y,z),
	the relation must contain (x,z).

Determine whether each relation is Transitive or not.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Transitive or not.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















List the properties each relation possesses.
All relations are defined on the set S = { 1, 2, 3, 4 }.

A = { (1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4) }

B = { (1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4) }