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Order 7 of the following sentences so that they form a logical proof of the statement:

Suppose $R$ is a symmetric and transitive relation on $A$ (i.e. $A \times A$ ) . Further suppose that for each $a \in A$ that there exists $b \in A$ such that $(a,b) \in R$ .

Show: $R$ is an equivalence relation.

Quick Hint? What makes $R$ an equivalence relation?

Choose from these statements.

$\exists b \in A$ such that $(a,b) \in R$
$R$ is an equivalence relation
Let $a$ be an arbitrary element of $A$ .
$(a,a) \in R \implies \exists b \in A$ such that $(a,b) \in R$ and $(b,a) \in R$ by transitivity
Too much may be the equivalent of none at all.
$R$ is reflexive
$(b,a) \in R$ by symmetry
Let $(a,b)$ be an arbitrary element of $R$ .
$(a,b) \in R$ and $(b,a) \in R$ implies $(a,a) \in R$ by transitivity
Assume $R$ is symmetric and transitive and $\forall a\in A, (a,a) \in R$ .
Assume that $R$ is symmetric and transitive on $A$ and that each element in $A$ is related to at least one other element in $A$ .

The proof statements in correct order.

Choose from these statements.

$\exists b \in A$ such that $(a,b) \in R$
$R$ is an equivalence relation
Let $a$ be an arbitrary element of $A$ .
$(a,a) \in R \implies \exists b \in A$ such that $(a,b) \in R$ and $(b,a) \in R$ by transitivity
Too much may be the equivalent of none at all.
$R$ is reflexive
$(b,a) \in R$ by symmetry
Let $(a,b)$ be an arbitrary element of $R$ .
$(a,b) \in R$ and $(b,a) \in R$ implies $(a,a) \in R$ by transitivity
Assume $R$ is symmetric and transitive and $\forall a\in A, (a,a) \in R$ .
Assume that $R$ is symmetric and transitive on $A$ and that each element in $A$ is related to at least one other element in $A$ .

The proof statements in correct order.