Mathematical Analysis I, 2004a

§§4–7. Relations. Mappings 9
Hence R, in turn, is the inverse of R −1 ; i.e.,
(R −1 ) −1 = R.
For example, the relations < and > between numbers are inverse to each other;
so also are the relations ⊆ and ⊇ between sets. (We may treat “⊆” as the name
of the set of all pairs (X, Y ) such that X ⊆ Y in a given space.)
If R contains the pairs (x, x ′ ), (y, y ′ ), (z, z ′ ), ..., we shall write
R =
(
x y z
x ′ y ′ z ′ ···
)
; e.g., R =
(
1 4 1 3
2 2 1 1
)
. (1)
To obtain R −1 , we simply interchange the upper and lower rows in (1).
Definition 1.
The set of all left terms x of pairs (x, y) ∈ R is called the domain of R,
denoted D R .Thesetofallright terms of these pairs is called the range
of R, denoted D R ′ . Clearly, x ∈ D R iff xRy for some y. In symbols,
x ∈ D R ⇐⇒ (∃ y) xRy; similarly, y ∈ D R ′ ⇐⇒ (∃ x) xRy.
In (1), D R is the upper row, and D ′ R
is the lower row. Clearly,
D R
−1 = D ′ R and D′ R −1 = D R.
For example, if
then
R =
( )
1 4 1
,
2 2 1
D R = D ′ R −1 = {1, 4} and D′ R = D R
−1 = {1, 2}.
Definition 2.
The image of a set A under a relation R (briefly, the R-image of A) is the
set of all R-relatives of elements of A, denoted R[A]. The inverse image
of A under R is the image of A under the inverse relation, i.e., R −1 [A].
If A consists of a single element, A = {x}, thenR[A] andR −1 [A] arealso
written R[x] andR −1 [x], respectively, instead of R[{x}] andR −1 [{x}].
Example.
Let
( )
1 1 1 2 2 3 3 3 3 7
R =
, A = {1, 2}, B= {2, 4}.
1 3 4 5 3 4 1 3 5 1