We have, $A=\left\{1,2,3,4,5,6,7\right\}$

Reflexive: A relation $R$ on a set $A$ is said to be reflexive if every element of $A$ is related to itself.

Thus, $R \mathrm{is} \mathrm{reflexive}\iff (a,a)\in R \mathrm{for} \mathrm{all} a\in A$

$\because (1,1),\left(2,2\right),(3,3),......(7,7)$ does not satisfy $x+y=7$

Hence $R$ is not reflexive.

Symmetric: A relation $\mathrm{R}$ is symmetric on a set $\mathrm{A}$ iff

$(a,b)\in R\Rightarrow (b,a)\in R \mathrm{for} \mathrm{all} a,b\in A$

$\Rightarrow x+y=7$

Now on interchanging $y$ and $x$ we get, $y+x=7$ which is always true for given set,

Hence $R$ is symmetric.

Transitive: A relation $R$ on $A$ is said to be transitive relation iff 

$\left(\mathit{a}\mathit{,}\mathit{b}\right)\mathit{\in }R\mathit{ }\mathrm{and}\mathit{ }\mathit{(}b\mathit{,}c\mathit{)}\mathit{\in }R$

$\Rightarrow \left(a,c\right)\in R \mathrm{for} \mathrm{all} a,b,c\in A$

Now taking $\left(a,b\right)\equiv \left(3,4\right)$ and $\left(b,c\right)\equiv \left(4,3\right)$ so  $\left(a,c\right)\equiv \left(3,3\right)$ does not satisfy $x+y=7$,

Hence, $R$ is not transitive and not equivalence.

Therefore, $R$ is only Symmetric.