How many symmetric relations have (x,y)

Question

Set $w = \{x,y,z,w\}$. How many symmetric relations contain $(x,y)$.

I know how to calculate the number of symmetric relations. But how do you calculate how many of those contain $(x,y)$?

Answer 1

HINT: Find a bijection between the set of symmetric relations on $\{x,y,z,w\}$ that do contain $\langle x,y\rangle$ and the set of symmetric relations on $\{x,y,z,w\}$ that don’t contain $\langle x,y\rangle$. Then use your knowledge of the total number of symmetric relations on $\{x,y,z,w\}$.