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: $A - B = \{x | x \in A \text{ and } x \notin B\}$
Complement: $A' = U - A$
De Morgan's Laws:
- $(A \cup B)' = A' \cap B'$
- $(A \cap B)' = A' \cup B'$
Cardinality: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$

Cartesian Product: $A \times B = \{(a,b) | a \in A, b \in B\}$
Relation: A subset of $A \times B$.
Types of Relations on a set $A$:
- Reflexive: $(a,a) \in R$ for all $a \in A$
- Symmetric: If $(a,b) \in R$, then $(b,a) \in R$
- Transitive: If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$
- Equivalence Relation: Reflexive, Symmetric, and Transitive

Definition: A relation $f: A \to B$ such that every element in $A$ has exactly one image in $B$.
Domain: Set of all possible input values ($A$).
Codomain: Set $B$.
Range: Set of all output values ($f(A) \subseteq B$).
Types of Functions:
- One-to-one (Injective): If $f(x_1) = f(x_2) \implies x_1 = x_2$.
- Onto (Surjective): For every $y \in B$, there exists $x \in A$ such that $f(x)=y$.
- Bijective: Both one-to-one and onto.
Composition of Functions: $(g \circ f)(x) = g(f(x))$
Inverse Function ($f^{-1}$): Exists if and only if $f$ is bijective. If $y=f(x)$, then $x=f^{-1}(y)$.
Algebra of Functions:
- $(f \pm g)(x) = f(x) \pm g(x)$
- $(fg)(x) = f(x)g(x)$
- $(f/g)(x) = f(x)/g(x), g(x) \ne 0$

Definition: $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that $0 < |x-a| < \delta \implies |f(x)-L| < \epsilon$.
Left Hand Limit (LHL): $\lim_{x \to a^-} f(x)$
Right Hand Limit (RHL): $\lim_{x \to a^+} f(x)$
Limit exists if and only if LHL = RHL = L.
Limit Laws: (Assuming $\lim f(x)$ and $\lim g(x)$ exist)
- $\lim (f(x) \pm g(x)) = \lim f(x) \pm \lim g(x)$
- $\lim (cf(x)) = c \lim f(x)$
- $\lim (f(x)g(x)) = \lim f(x) \lim g(x)$
- $\lim (f(x)/g(x)) = \lim f(x) / \lim g(x)$, if $\lim g(x) \ne 0$
Standard Limits:
- $\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$
- $\lim_{x \to 0} \frac{\tan x}{x} = 1$
- $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
- $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
- $\lim_{x \to 0} \frac{\log(1+x)}{x} = 1$
- $\lim_{x \to 0} (1+x)^{1/x} = e$
- $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$
L'Hôpital's Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ is of form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).