Just pass false for the inherited argument of public_methods:
"hello".public_methods.include?(:dup) # => true
"hello".public_methods(false).include?(:dup) # => false
Not an answer to your question, but in case you didn’t know, irb does autocompletion, so it’s easy to get the list of public methods (especially if you know the beginning of the method you are looking for). Just hit tab; hitting it twice will list all possibilities (including inherited ones, though):
> "nice".d<tab><tab>
"nice".delete "nice".delete! "nice".display "nice".downcase
"nice".downcase! "nice".dump "nice".dup "nice".define_singleton_method
> "nice".<tab><tab>
Display all 162 possibilities? (y or n)
...
Using pry makes it even easier to see the methods available, broken down by level of inheritance:
[1] pry(main)> cd "nice"
[2] pry("nice"):1> ls
Comparable#methods: < <= > >= between?
String#methods: % * + << <=> == === =~ [] []= ascii_only? bytes bytesize byteslice capitalize capitalize! casecmp center chars chomp chomp! chop chop! chr clear codepoints concat count crypt delete delete! downcase downcase! dump each_byte each_char each_codepoint each_line empty? encode encode! encoding end_with? eql? force_encoding getbyte gsub gsub! hash hex include? index insert inspect intern length lines ljust lstrip lstrip! match next next! oct ord partition prepend replace reverse reverse! rindex rjust rpartition rstrip rstrip! scan setbyte shellescape shellsplit size slice slice! split squeeze squeeze! start_with? strip strip! sub sub! succ succ! sum swapcase swapcase! to_c to_f to_i to_r to_s to_str to_sym tr tr! tr_s tr_s! unpack upcase upcase! upto valid_encoding?
locals: _ _dir_ _ex_ _file_ _in_ _out_ _pry_