To illustrate our idea, we first show that in the extreme case when word vectors we try to learn are binary vectors precisely representing set memberships, the semantic relation between two words can be determined by comparing the two vectors element-wise.
let D denote the size of the universe D. Since a word x can be regarded as a subset of D, we assume that the entailment vector of x is a D-dimensional binary vector x, where x i is 1 if the i th element in D is inside the set x and 0 otherwise.