Question

Let A={1,2,3}
a.) R is the relation < on A
b.) R is the relation >= on A
c. ) R is the relation ⊂ on Ƥ (A)

Answer 1

Question 1.

Let $A=\{1,2,3\}$. Describe $R$ if $R$ is the following relation:

1. $R$ is the relation $<$ on $A$;

2. $R$ is the relation $\geq$ on $A$;

3. $R$ is the relation $\subset$ on $P(A)$.

Solution.

(a) $R$ is the relation “to be strictly less than”, i. e. $(a,b)\in R$ if and only if $a$ is strictly less that $b$. We can even write all the pairs $(a,b)$, which belong to $R$:

$R=\{(1,2),(1,3),(2,3)\}.$

However, for example, $(1,1)\not\in R$, because 1 equals 1 and hence it is not strictly less that 1.

(b) $R$ is the relation “to be greater than or equal to”, i. e. $(a,b)\in R$ if and only if $a$ is strictly greater than or equal to $b$. Equivalently, $(a,b)\in R$ if and only if $a$ is not strictly less than $b$. We see that this relation is the complement of the relation $<$ considered above. So,

$R=\{(1,1),(2,1),(2,2),(3,1),(3,2),(3,3)\}.$

(c) 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. ∎