Mathematics: Sets, Relations, Trigonometric Functions, Induction, Complex Numbers, and Inequalities

Flashcards covering key definitions, mathematical rules, formulas, and theorems from chapters on Sets, Relations, Trigonometry, Induction, Complex Numbers, Inequalities, and Counting.

Who developed the theory of sets, and while working on what type of problems did he first encounter them?

German mathematician Georg Cantor developed the theory of sets, and he first encountered them while working on "problems on trigonometric series".

In roster form, how is the set of all natural numbers that divide $42$ represented?

$$\{1, 2, 3, 6, 7, 14, 21, 42\}$$

What is the definition of an empty set, and what symbols are used to denote it?

A set which does not contain any element is called the empty set, null set, or void set. It is denoted by the symbol $\phi$ or $\{\}$.

If a set $A$ has $n(A) = m$ elements, how many total elements are in its power set $P(A)$?

$$2^m$$

What are De Morgan's laws for two subsets $A$ and $B$ of a universal set $U$?

$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$

Given two non-empty sets $P$ and $Q$, how is the Cartesian product $P \times Q$ defined?

$$P \times Q = \{(p, q) : p \in P, q \in Q\}$$

What is the definition of a relation $R$ from a non-empty set $A$ to a non-empty set $B$?

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the Cartesian product $A \times B$.

If $n(A) = p$ and $n(B) = q$, what is the total number of relations that can be defined from set $A$ to set $B$?

$$2^{pq}$$

What is the relation between degree measure and radian measure for a complete revolution of a circle?

$2\pi\,\text{radians} = 360^\circ$ (or $\pi\,\text{radians} = 180^\circ$)

What formula relates arc length $l$, radius $r$, and central angle $\theta$ in radians in a circle?

$\theta = \frac{l}{r}$ or $l = r \times \theta$

What is the general solution for $x$ when $\sin(x) = \sin(y)$?

$x = n\pi + (-1)^n y$, where $n \in \mathbb{Z}$

What is the expansion identity for $\cos(x + y)$ in terms of sine and cosine functions?

$$\cos(x + y) = \cos(x) \times \cos(y) - \sin(x) \times \sin(y)$$

What two conditions must be satisfied to prove a statement $P(n)$ for all natural numbers $n$ using the Principle of Mathematical Induction?

(i) The statement is true for $n = 1$, i.e., $P(1)$ is true. (ii) If the statement is true for $n = k$ (where $k$ is a positive integer), then the statement is also true for $n = k + 1$.

For a complex number $z = a + \text{i}b$, how are its real part $\text{Re}(z)$ and imaginary part $\text{Im}(z)$ defined?

$\text{Re}(z) = a$ and $\text{Im}(z) = b$

What is the formula for the multiplicative inverse $z^{-1}$ of a non-zero complex number $z = a + \text{i}b$?

$$z^{-1} = \frac{\bar{z}}{|z|^2} = \frac{a}{a^2 + b^2} + \text{i}\frac{-b}{a^2 + b^2}$$

What is the polar representation of a non-zero complex number $z = x + \text{i}y$?

$z = r(\cos(\theta) + \text{i}\sin(\theta))$, where $r = \sqrt{x^2 + y^2}$ is the modulus and $\theta$ is the argument.

What is the Fundamental Theorem of Algebra?

A polynomial equation has at least one root.

What happens to the sign of an inequality when both sides are multiplied or divided by a negative number?

The sign of inequality is reversed.

What is the difference between a strict inequality and a slack inequality?

Inequalities involving '$<$' or '$>$' are strict inequalities, whereas inequalities involving '$\le$' or '$\ge$' are slack inequalities.

How many total pairs of a pant and a shirt can Mohan dress up with if he has $3$ pants and $2$ shirts?

$3 \times 2 = 6$ pairs