Sets and Vectors (Basics for Economics)

1) Sets: core definitions

A set is a well-defined collection of distinct objects (elements).

1.1 Notation

• Membership: $x \in A$, $x \notin A$
• Empty set: $\varnothing$
• Universal set: $U$

1.2 Types of sets

• Finite (limited elements), Infinite
• Subset: $A \subseteq B$
• Proper subset: $A \subset B$
• Equal sets: $A = B$ if they have same elements

1.3 Set operations (very testable)

• Union: $A \cup B = \{x: x \in A \text{ or } x \in B\}$
• Intersection: $A \cap B = \{x: x \in A \text{ and } x \in B\}$
• Difference: $A \setminus B = \{x: x \in A, x \notin B\}$
• Complement: $A^c = U \setminus A$

1.4 Laws (high-yield identities)

• Commutative: $A\cup B=B\cup A$, $A\cap B=B\cap A$
• Associative: $(A\cup B)\cup C=A\cup(B\cup C)$ (same for ∩)
• Distributive: $$A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$$ $$A\cup(B\cap C)=(A\cup B)\cap(A\cup C)$$
• De Morgan’s laws: $$(A\cup B)^c=A^c\cap B^c,\quad (A\cap B)^c=A^c\cup B^c$$

1.5 Counting (basic)

If A and B are finite:

Used in probability/survey data type questions.

2) Relations and functions (bridge concept)

A relation from set $A$ to set $B$ is a subset of $A\times B$ (Cartesian product).