Question

Let A ={1,2,3}
what is R if R is the relation ⊂ Ƥ (A)

How can you say that a relation is complete?
Complete is a property of relation.
please give me an example.

Thank you!

Answer 1

Question 1.

Let $A=\{1,2,3\}$. What is $R$ if $R$ is the relation $\subset$ on $P(A)$. How can you say that a relation is complete?

Solution. Recall that $P(A)$ is the set of all subsets of $A=\{1,2,3\}$, i. e.

$P(A)=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}.$

The relation $\subset$ on $P(A)$ is the relation “to be a proper subset”. This means that for all $X,Y\in P(A)$ we have $X\subset Y$ if and only if $X$ is a subset of $Y$, and there are elements of $Y$ which do not belong to $X$. For example, $\{1\}\subset\{1,3\}$, because $\{1\}$ is contained in $\{1,3\}$ and does not coincide with the whole $\{1,3\}$. But $\{1\}\not\subset\{2,3\}$, because $\{1\}$ is not a subset of $\{2,3\}$. Moreover, $\{1\}\not\subset\{1\}$, because $\{1\}$ is a subset of $\{1\}$, which is not proper.

A relation $R$ on a set $A$ is said to be complete if $R=A\times A$, i. e. for all $a,b\in A$ we have $(a,b)\in R$. It is the maximal possible relation on $A$ in the sense that any relation on $A$ is contained in $R$. $\Box$