Chapter 1.7, Problem 1TFE

Label each of the following statements as either true or false.

Every mapping on a nonempty set $A$ is a relation.

To determine

Whether the statement, “Every mapping on the nonempty set $A$ is a relation” is true or false.

Answer to Problem 1TFE

Solution:

The statement, “Every mapping on the nonempty set $A$ is a relation” is true.

Explanation of Solution

Consider the statement, “Every mapping on the nonempty set $A$ is a relation”.

Let $A$ and $B$ are non-empty sets. A subset $f$ of $A\times B$ is a mapping from $A$ to $B$ if and only if for every $a\in A$ there exists unique(one and only one) image $b\in B$ such that $\left(a,b\right)\in f$

A relation on a nonempty set $A$ is a nonempty set $R$ of ordered pairs $\left(x,y\right)$ of elements of $x$ and $y$ of $A$.

From the above definitions,

Every mapping is a relation.

Hence the statement, “Every mapping on the nonempty set $A$ is a relation” is true.