Sets, Functions and Continuity, Sequence, Series | UGC NET Economics – Notes

Sets

A set is a well-defined collection of distinct objects, considered as an object in its own right. The objects that make up a set are called its elements or members. If (x) is an element of a set (A), this is denoted \(x \in A\); if not, \(x \notin A\). Sets form the foundational language of modern mathematical economics, since every economic model—whether describing a consumer’s budget set, a producer’s production possibility set, or the choice set available to an agent—is built upon set-theoretic notation and reasoning.

Sets can be described in two principal ways. The roster method (or tabular form) lists all elements explicitly, such as \(A = {1, 2, 3, 4, 5}\). The set-builder method (or rule method) specifies a defining property, such as \(A = {x : x \text{ is a natural number}, x \leq 5}\). The set-builder notation is especially important in economics because economic constraint sets are almost always expressed this way—for instance, a budget set is written as $$B = {(x_1, x_2) : p_1 x_1 + p_2 x_2 \leq m,\ x_1, x_2 \geq 0}$$ where \(p_1, p_2\) are prices and (m) is income.

Types of Sets

A set containing no elements is called the empty set or null set, denoted \(\emptyset) or ({}\). A singleton set contains exactly one element. A finite set has a countable, limited number of elements, while an infinite set has an unlimited number of elements, such as the set of natural numbers \(\mathbb{N}\). A universal set, denoted (U), is the set containing all objects under consideration in a particular context.

The concept of a subset is central: a set (A) is a subset of set (B), written \(A \subseteq B\), if every element of (A) is also an element of (B). If additionally \(A \neq B\), then (A) is called a proper subset of (B), written \(A \subset B\). Two sets are said to be equal if they contain exactly the same elements, regardless of order or repetition. The power set of a set (A), denoted \(P(A)\), is the set of all possible subsets of (A), including the empty set and (A) itself; if (A) has (n) elements, then \(P(A)\) has \(2^n\) elements, expressed as \(|P(A)| = 2^n\).

Set Operations

The union of two sets (A) and (B), denoted \(A \cup B\), is the set of elements belonging to (A) or (B) or both: $$A \cup B = {x : x \in A \text{ or } x \in B}$$

The intersection of (A) and (B), denoted (A \cap B), is the set of elements common to both: $$A \cap B = {x : x \in A \text{ and } x \in B}$$

If \(A \cap B = \emptyset\), the sets are called disjoint sets. The difference of two sets, (A – B) (or \(A \setminus B)\), is the set of elements in (A) but not in (B). The complement of a set (A) with respect to the universal set (U), denoted \(A^c\) or \(A’\), consists of all elements in \(U\) not in \(A\): \(A^c = U – A\).

Set operations obey important algebraic laws analogous to arithmetic, collectively known as the algebra of sets. These include the commutative laws \((A \cup B = B \cup A); (A \cap B = B \cap A)\), the associative laws, the distributive laws \((A \cup (B \cap C) = (A \cup B) \cap (A \cup C))\), and De Morgan’s Laws, which state: $$(A \cup B)^c = A^c \cap B^c \qquad (A \cap B)^c = A^c \cup B^c$$

For counting purposes, the cardinality of a finite set (A), denoted (n(A)) or (|A|), is the number of elements it contains. For two sets, the fundamental counting formula is: $$n(A \cup B) = n(A) + n(B) – n(A \cap B)$$

For example, if a survey of 100 consumers finds that 60 buy tea, 45 buy coffee, and 25 buy both, then the number buying at least one beverage is \(n(A \cup B) = 60 + 45 – 25 = 80\).