elements; a roster involving an ellipsis would be an example. Membership If is a set and is an element of , this is written in shorthand as , which can also be read as "x belongs to B", or "x is in B". The statement "y is not an element of B" is written as , which can also be read as "y is not in B". For example, with respect to the sets , , and , The empty set The empty set (or null set) is the unique set that has no members. It is denoted or or or (or ). Singleton sets A singleton set is a set with exactly one element; such a set may also be called a unit set. Any such set can be written as , where x is the element. The set and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat. Subsets If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written A ⊆ B, or B ⊇ A. The latter notation may be read B contains A, B includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B. If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A. A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset), while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B. Examples: The set of all humans is a proper subset of the set of all mammals. ⊂ . ⊆ . The empty set is a subset of every set, and every set is a subset of itself: ∅ ⊆ A.