Certain facts. Provable and otherwise about cookies.

Mac states it is nec­es­sary, but suf­fi­cient, that his cook­ie be Big.

For Scar­lett, it is nec­es­sary and suf­fi­cient that she have a cookie.

For Mac, it is sufficient—but not necessary—that Mac and Scar­lett each have a cook­ie(cf.1).

For Scar­lett, it is sufficient—but not necessary—that Scar­lett and Mac each have a cook­ie(cf.1).

State­ments #3 and #4 are not iden­ti­cal. The proof is left for the reader.