1.3.6 Special Sets

Certain sets are of particular significance and are referred to as special sets. The two key special sets frequently used in defining and calculating probabilities are the empty set and the universal set.

An empty set is a set that contains no elements, denoted by the symbol of a slashed zero  or { }. It's a subset of every set i.e. If  is set, then  ⊂ . The cardinality of an empty set is 0, signifying that there are no elements within it.

A set containing element 0 is not an empty set, it’s a set containing element 0, denoted as {0}. The cardinality of this set is 1.

Alternatively, a set of all possible elements we could consider within the context of our study is called a universal set. In other words, the universal set contains every element that is relevant to our question. It is typically denoted by the symbol  or sometimes by the specific letter or symbol that best represents the context of the problem. For example, in the context of flipping a coin, our universal set may be defined as  = {Heads, Tails}, or in the context of rolling a die, our universal set may be defined as  = {1, 2, 3, 4, 5, 6}. As is clear from the definition, the universal set is context-dependent and changes as our context changes.