Mathematics I Flashcards

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Week 1: Sets & Functions

• Power Set cardinality of set A with $n$ elements :: $2^n$
• De Morgan's Law for $(A \cup B)'$ :: $A' \cap B'$
• Relation that is Reflexive, Symmetric, and Transitive :: Equivalence Relation
• Injection condition for $x_1, x_2$ :: $f(x_1) = f(x_2) \implies x_1 = x_2$
• Surjection condition :: Range = Co-domain
• Subset of $A \times B$ :: Relation
• Cardinality of $\emptyset$ :: $0$

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Week 2: Coordinate Geometry

• Centroid of triangle with vertices $(x1, y1), (x2, y2), (x3, y3)$ :: $(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$
• Slope of a line perpendicular to $m$ :: $-1/m$
• Distance of point $(x1, y1)$ from line $ax + by + c = 0$ :: $\frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$
• Distance between parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$ :: $\frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$
• Shoelace formula area result sign :: Always use Absolute value
• Vertical line slope :: Undefined (Division by 0)

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Week 3: Quadratics

• Vertex of $ax^2 + bx + c$ :: $x = -b / (2a)$
• Discriminant $D$ for unique real root :: $D = 0$
• Sum of roots of $ax^2 + bx + c$ :: $-b/a$
• Product of roots of $ax^2 + bx + c$ :: $c/a$
• Concavity of parabola if $a < 0$ :: Downwards (Max value at vertex)
• Roots of $ax^2 + bx + c = 0$ :: $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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Week 4: Polynomials

• Max turning points of degree $n$ polynomial :: $n - 1$
• Behavior of $p(x)$ with leading term $-x^5$ as $x \to \infty$ :: $-\infty$
• Multiplicity $m$ is even :: Graph touches/tangent to X-axis
• Multiplicity $m$ is odd :: Graph crosses X-axis
• Definition of a root :: Value of $x$ where $p(x) = 0$
• Remainder Theorem: $P(x)$ divided by $(x-a)$ :: Remainder exists as $P(a)$

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