Q21Discrete Mathematics

Set theory forms the absolute axiomatic foundation of modern mathematics. A "Set" is strictly defined as a well-defined, unordered mathematical collection of distinct objects (elements). To systematically analyze these collections, mathematics categorizes sets into several highly specific types based on their cardinality (size), content, and comparative relationships.

1. Cardinality-Based Classifications

• Finite Set: A set is mathematically finite if the total count of its constituent elements is a specific, known, non-negative integer. The process of counting the elements will definitively terminate. Example: $A=\{2,4,6,8\}$. The cardinality $|A|=4$.
• Infinite Set: A set is infinite if it is not finite. Its elements cannot be exhaustively listed or counted; the process would continue endlessly. Example: The set of all Natural Numbers, $N=\{1,2,3,\dots \}$.
• Empty Set (Null or Void Set): This is a highly specialized finite set that physically contains absolutely zero elements. Its cardinality is strictly 0. It is denoted universally by the symbol $\varnothing$ or an empty pair of braces $\{\}$. Crucially, it is mathematically considered a subset of every single other set in existence.
• Singleton Set: A finite set that contains exactly one single, solitary element. Its cardinality is exactly 1. Example: $B=\{7\}$.

2. Relational Classifications

• Equal Sets: Two sets, $A$ and $B$, are declared strictly "Equal" ($A=B$) if and only if they contain the exact, identical specific elements. Order and repetition are mathematically irrelevant. Example: If $A=\{1,2,3\}$ and $B=\{3,2,1,1\}$, then $A=B$. Every element in $A$ is in $B$, and vice versa.
• Equivalent Sets: Two sets, $A$ and $B$, are declared "Equivalent" (denoted $A\sim B$) if they possess the exact same cardinality (total number of elements), regardless of what those specific elements actually are. Example: If $A=\{x,y,z\}$ and $B=\{4,5,6\}$, they are equivalent because $|A|=3$ and $|B|=3$. Equal sets are always equivalent, but equivalent sets are rarely equal.

3. Structural Classifications

• Universal Set: Denoted usually by $U$, this is the supreme, overarching "master set" within the context of a specific mathematical problem. It explicitly contains every single element, object, or entity that is currently under consideration or discussion.
• Power Set: The Power Set of any set $S$, denoted as $P(S)$, is a "set of sets." It is mathematically defined as the complete collection of all possible subsets that can be generated from $S$, explicitly including the empty set $\varnothing$ and the original set $S$ itself. If a set $S$ possesses $n$ elements, its corresponding Power Set is mathematically guaranteed to possess exactly $2^n$ subsets. Example: If $S=\{1,2\}$, then $P(S)=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$.