EE 234b Formula Sheet: Sets, Sequences, Recurrence, and Summation

Practice flashcards covering definitions, set laws, counting principles, sequence formulas, recurrence relations, and summation rules from the EE 234b formula sheet.

Union of Sets ($A \cup B$)

The set of all elements that belong to $A$ or $B$, defined as $\{ x \mid x \in A \text{ or } x \in B \}$.

Intersection of Sets ($A \cap B$)

The set of all elements that belong to both $A$ and $B$, defined as $\{ x \mid x \in A \text{ and } x \in B \}$.

Set Difference ($A \setminus B$)

The set of elements belonging to $A$ but not to $B$, defined as $\{ x \mid x \in A \text{ and } x \notin B \}$, which is equivalent to $A \cap B'$.

Set Complement ($A'$)

The set of all elements in the universal set that do not belong to $A$, defined as $\{ x \mid x \notin A \}$.

Disjoint Sets

Two sets $A$ and $B$ that share no elements, formally defined by $A \cap B = \emptyset$.

Distributive Laws of Sets

The set identities stating $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

De Morgan's Laws (Sets)

The set laws stating that the complement of a union is the intersection of complements, $(A \cup B)' = A' \cap B'$, and the complement of an intersection is the union of complements, $(A \cap B)' = A' \cup B'$.

Symmetric Difference of Sets

The set consisting of elements in either set but not in both, expressed as $(A \setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A \cap B)$.

Inclusion-Exclusion Principle (Two Sets)

The formula for the cardinality of the union of two sets: $|A \cup B| = |A| + |B| - |A \cap B|$.

Inclusion-Exclusion Principle (Three Sets)

The formula for the cardinality of the union of three sets: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.

$n\text{th}$ Term of an Arithmetic Sequence

The formula for the $n\text{th}$ term given by $a_n = a_1 + (n - 1)d$, where the common difference is $d = a_2 - a_1$.

Sum of an Arithmetic Sequence ($S_n$)

The sum of the first $n$ terms given by $S_n = \frac{n}{2}(a_1 + a_n)$, or $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$ when the last term is unknown.

$n\text{th}$ Term of a Geometric Sequence

The formula for the $n\text{th}$ term given by $a_n = a r^{n-1}$, where the common ratio is $r = \frac{a_2}{a_1}$.

Sum of a Finite Geometric Sequence ($S_n$)

The sum of the first $n$ terms given by $S_n = \frac{a(1 - r^n)}{1 - r}$, where $r \neq 1$.

Sum of an Infinite Geometric Series ($S_\infty$)

The infinite sum given by $S_\infty = \frac{a}{1 - r}$, which is valid when $|r| < 1$.

Arithmetic Recurrence Relation

A recurrence of the form $a_n = a_{n-1} + d$, with closed-form solution $a_n = a_0 + nd$.

Geometric Recurrence Relation

A recurrence of the form $a_n = r a_{n-1}$, with closed-form solution $a_n = a_0 r^n$.

Fibonacci Recurrence Relation

A recurrence defined by $a_n = a_{n-1} + a_{n-2}$ with initial conditions $a_0 = 0$ and $a_1 = 1$.

Unrolling Method

A technique for solving recurrence relations by expanding successive terms ($a_1, a_2, a_3, \dots$), spotting the pattern, and writing $a_n$ using $n$ only.

Number of Terms in a Summation

In sigma notation $\sum_{i=m}^n a_i$, the total number of terms summed is given by $n - m + 1$.

Sum of a Constant Formula

The summation formula for a constant $c$ given by $\sum_{i=1}^n c = c n$.

Sum of Integers Formula

The summation formula for the first $n$ positive integers: $\sum_{i=1}^n i = \frac{n(n + 1)}{2}$.

Sum of Squares Formula

The summation formula for the first $n$ squares: $\sum_{i=1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$.

Series Expansion of $\sin(x)$

The Taylor/power series expansion given by $\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n + 1)!}$.