Mathematical Reasoning: Sets, Subsets, Algebra of Sets

Mathematical Reasoning: Sets, Subsets, Algebra of Sets

1. Sets

A set is a well-defined collection of distinct objects, called elements or members of the set. Sets are usually denoted by capital letters such as $A, B, C$, and elements by lowercase letters such as $a, b, c$. If an element $a$ is in a set $A$, it is written as $a \in A$. If not, $a \notin A$.

There are several ways to describe a set:

• Roster (Tabular) form: Listing all elements, e.g., $A = \{1, 2, 3, 4\}$
• Set-builder form: Describing elements by a property, e.g., $A = \{x \mid x \text{ is a natural number less than 5} \}$

Special types of sets:

• Empty Set ($\emptyset$): A set with no elements.
• Finite Set: A set with a finite number of elements.
• Infinite Set: A set with an unending number of elements.
• Universal Set ($U$): The set that contains all possible elements under consideration.
• Power Set: The set of all subsets of a set $A$, denoted $P(A)$. If $A$ has $n$ elements, $P(A)$ has $2^n$ elements.

2. Subsets

If every element of set $A$ is also in set $B$, then $A$ is a subset of $B$, denoted $A \subseteq B$. If $A \subseteq B$ but $A \neq B$, then $A$ is a proper subset of $B$, written $A \subset B$.

Basic properties of subsets:

• Every set is a subset of itself: $A \subseteq A$
• The empty set is a subset of every set: $\emptyset \subseteq A$
• If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$ (transitivity)

3. Algebra of Sets

Operations on sets follow specific algebraic rules. The major operations include:

Union ( $A \cup B$ )

The union of sets $A$ and $B$ is the set containing all elements that are in $A$, in $B$, or in both.

Intersection ( $A \cap B$ )

The intersection of sets $A$ and $B$ contains only those elements that are common to both.

Difference ( $A - B$ or $A \setminus B$ )

The difference between sets $A$ and $B$ is the set of elements that are in $A$ but not in $B$.

Complement ( $A^c$ or $\overline{A}$ )

The complement of a set $A$ with respect to the universal set $U$ is the set of all elements in $U$ that are not in $A$.

Laws of Set Algebra

1. Idempotent Laws:

   • $A \cup A = A$
   • $A \cap A = A$

2. Identity Laws:

   • $A \cup \emptyset = A$
   • $A \cap U = A$

3. Domination Laws:

   • $A \cup U = U$
   • $A \cap \emptyset = \emptyset$

4. Complement Laws:

   • $A \cup A^c = U$
   • $A \cap A^c = \emptyset$

5. Commutative Laws:

   • $A \cup B = B \cup A$
   • $A \cap B = B \cap A$

6. Associative Laws:

   • $A \cup (B \cup C) = (A \cup B) \cup C$
   • $A \cap (B \cap C) = (A \cap B) \cap C$

7. Distributive Laws:

   • $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
   • $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

8. De Morgan’s Laws:

   • $(A \cup B)^c = A^c \cap B^c$
   • $(A \cap B)^c = A^c \cup B^c$

9. Double Complement Law:

   • $(A^c)^c = A$

10. Absorption Laws:

    • $A \cup (A \cap B) = A$
    • $A \cap (A \cup B) = A$