Relations - 1

Cartesian Product

For the Cartesian product you are making a list of all possibilities of the elements in both sets. This is similar to multiplying brackets.

Let $A=\{1,2\}$ and $B=\{a,b,c\}$, then:

\[A\times B = \{(1,a),(2,a),(1,b),(2,b),(1,c),(2,c)\}\]

Therefore:

\[B\times A = \{(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)\}\]

Any relation between the elements in set $A$ and $B$ will be in the set of their Cartesian product.

A binary relation between two sets $A$ and $B$ is a subset $R$ of the Cartesian product $A\times B$.

If $A=B$, then $R$ is called a binary relation on $A$.

The set $A$ is the set of all people in the tree.

graph TD
fm[Fred and Mavis] --- Alice
fm --- ks[Ken and Sue]
ks --- Jane
ks --- Fiona
ks --- Alan
jm[John and Mary] --- ks
jm --- Mike
jm --- Penny

$R=\{(x,y)\vert x\text{ is a grandfather of } y\}$

For this set:
\[R=\{\text{(Fred, Jane), (Fred, Fiona),}\text{ (Fred, Alan), (John, Jayne),}\text{ (John, Fiona), (John, Alan)}\}\]
$S=\{(x,y)\vert x\text{ is a sister of } y\}$

For this set:
\[S=\{\text{(Alice, Ken), (Sue, Mike),}\text{ (Sue, Penny), (Penny, Sue),}\text{ (Penny, Mike), (Jane, Fiona)}\}\]

Write down the ordered parts belonging to the following binary relations between $A=\{1,3,5,7\}$ and $B=\{2,4,6\}:$

$U=\{(x,y)\in A\times B \vert x + y = 9\}$

This means the combinations from the two sets where the elements sum to 9.

$U=\{(3,6),(5,4),(7,2)\}$
$V=\{(x,y)\in A\times B \vert x < y \}$

This is the set of all pairs such that the first element is smaller than the second element:
\[V=\{(1,2),(1,4),(1,6),(3,4),(3,6),(5,6)\}\]