Chapter 3.14, Problem 14.17E

Drawing pictures of relations. Pictures of mathematical objects are wonderful aids in understanding concepts. There is a nice way to draw a picture of a relation on a set or of a relation from one set to another.

To draw a picture of a relation R on a set A, we make a diagram in which each element of A is represented by a dot. If aRb, then we draw an arrow from dot a to dot b. If it should happen that b is also related to a, we draw another arrow from b to a. And if a R a, then we draw a looping arrow from a to itself.

For example, let $A=\left\{1,2,3,4,5\right\}$ and $R=\left\{\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(4,3\right),\left(3,1\right)\right\}$.

A picture of the relation R on A is given in the first figure. A picture of the relation R on A is given in the first figure.

To draw a picture of a relation from A to B, we draw two collections of dots. The first collection of dots corresponds to the elements in A, and we place these on the left side of the figure. The dots for B go on the right. We then draw an arrow from $a\in A$ to $b\in B$ just when $\left(a,b\right)$ is in the relation.

For example, let $A=\left\{1,2,3,4,5\right\}$ and $B=\left\{4,5,6,7\right\}$ and let S be the relation $\left\{\left(1,4\right),\left(1,5\right),\left(2,5\right),\left(3,6\right)\right\}$.

A picture of the relation S is given in the second figure.

Please draw pictures of the following relations.

1. Let and let R be the relation | (divides) restricted to A.
2. Let $A=\left\{1,2,3,4,5\right\}$ and let R be the less-than relation restricted to A.
3. Let $A=\left\{1,2,3,4,5\right\}$ and let R be the relation = (equals) restricted to A.
4. Let $A=\left\{1,2,3,4,5\right\}$ and let B D f2; 3; 4; 5g. Let R be the relation $\ge$ (greater than or equal to) from A to B.
5. Let $A=\left\{-1,-2,-3,-4,-5\right\}$ and let $B=\left\{1,2,3,4,5\right\}$ and let $R=\left\{\left(a,b\right):a\in A,b\in B,a\left|b\right.\right\}$.