How to write a robust memberQ that test membership in set theory sense?

Question

Select[{{"foo", "bar"} -> "a", {"foo", "baz"} -> 
   "a", {"foo", _} -> "a"}, MemberQ[{{"foo", "bar"}}, #[[1]]] &]

(* {{"foo", "bar"} -> "a", {"foo", _} -> "a"} *)

I'd like the answer to be {{"foo", "bar"} -> "a"}. The built-in MemberQ seems to use MatchQ for testing equality. I'd like a memberQ that uses SameQ or TrueQ[Equal[#1, #2]]&, and works for possibly funky arguments such as patterns.

Answer 1

If I understand the question you are essentially displeased (for your application) that:

MemberQ[{x[1, 2], x[3, 4]}, x[1, _]]

True

What you need is Verbatim:

MemberQ[{x[1, 2], x[3, 4]}, Verbatim[ x[1, _] ]]

False

Select[
  {{"foo", "bar"} -> "a", {"foo", "baz"} -> "a", {"foo", _} -> "a"},
  MemberQ[{{"foo", "bar"}}, Verbatim @ #[[1]]] &
]

{{"foo", "bar"} -> "a"}

In case the behavior of Equal is necessary or desirable you can use that instead like this:

Select[
  {{"foo", 0.5} -> "a", {"foo", "baz"} -> "a", {"foo", _} -> "a"},
  MemberQ[{{"foo", 1/2}}, x_ /; x == #[[1]] ] &
]

{{"foo", 0.5} -> "a"}

Note 0.5 and 1/2.