Hi John.

A couple of small typos:

\$\$ x \sim_{f^{\ast}(P \wedge Q)} x' \textrm{ if and only if } x \sim_{ f^{\ast}(P)} x' \textrm{ and } x_{f^{\ast}(Q) } x'. \$\$

should be:

\$\$ x \sim_{f^{\ast}(P \wedge Q)} x' \textrm{ if and only if } x \sim_{ f^{\ast}(P)} x' \textrm{ and } x \sim_{f^{\ast}(Q) } x'. \$\$

The next equation also has a small typo and should be:

\$\$ f(x) \sim_{P \wedge Q} f(x') \textrm{ if and only if } f(x) \sim_P f(x') \textrm{ and } f(x) \sim_Q f(x'). \$\$

Also I think the join \\(P \vee Q \\) has two parts:

\$\$ P \vee Q = \\{\\{11,12,13,22,23\\},\\{21\\}\\} . \$\$