Algebra and Trigonometry

These flashcards cover key vocabulary and definitions on Algebra and Trigonometry concepts discussed in lecture notes.

Set

A well-defined collection of distinct objects, known as elements.

Union of sets (A ∪ B)

The set of elements which belong to either A or B or both.

Intersection of sets (A ∩ B)

The set of elements that belong to both A and B.

Subset (A ⊆ B)

A set A is a subset of a set B if every member of A is a member of B.

Proper subset (A ⊂ B)

A subset A of B is called a proper subset if A is not equal to B.

Complement of a set (A′)

The set of all elements not in the set A relative to a universal set S.

Disjoint sets

Two sets A and B are disjoint if they have no elements in common.

Cartesian Product (S × T)

The set of all possible ordered pairs (s, t) where s ∈ S and t ∈ T.

Polynomial function

A function defined by a polynomial of the form P(x) = anx^n + an-1x^(n-1) + … + a1x + a0.

Real Numbers

Set of all types of numbers excluding complex numbers, including natural, whole, integers, rational, and irrational.

Rational Numbers (Q)

Numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

Irrational Numbers (Q′)

Numbers that cannot be expressed as a fraction of two integers.

Absolute Value |a|

The distance between the point ‘a’ and the origin, defined as |a| = a if a ≥ 0, -a if a < 0.

Vertical Asymptote

A vertical line x = a where the function f(x) approaches positive or negative infinity.

Horizontal Asymptote

A horizontal line y = b where the function f(x) approaches the value b as x approaches infinity.

Zeros of a Function

The values of x for which f(x) = 0.

Factor Theorem

If p is a zero of f(x), then (x - p) is a factor of f(x).

Quadratic Formula

The solutions of the quadratic equation ax² + bx + c = 0 are given by x = (-b ± √(b² - 4ac)) / 2a.

Venn Diagram

A diagram that shows all possible logical relations between a finite collection of sets.

Radical Function

A function that contains a root expression, such as √x.

Set

A well-defined collection of distinct objects, known as elements.

Union of sets (A \cup B)

The set of elements which belong to either A or B or both.

Intersection of sets (A \cap B)

The set of elements that belong to both A and B.

Subset (A \subseteq B)

A set A is a subset of a set B if every member of A is a member of B.

Proper subset (A \subset B)

A subset A of B is called a proper subset if A is not equal to B.

Complement of a set (A' or A^c)

The set of all elements not in the set A relative to a universal set S.

Disjoint sets

Two sets A and B are disjoint if they have no elements in common.

Cartesian Product (S \times T)

The set of all possible ordered pairs (s, t) where s \in S and t \in T.

Polynomial function

A function defined by a polynomial of the form P(x) = an x^n + a{n-1} x^{n-1} + \dots + a1 x + a0.

Real Numbers (R)

Set of all types of numbers excluding complex numbers, including natural, whole, integers, rational, and irrational.

Rational Numbers (Q)

Numbers that can be expressed as the quotient of two integers, \frac{p}{q}, where q \neq 0.

Irrational Numbers (Q')

Numbers that cannot be expressed as a fraction of two integers.

Absolute Value |a|

The distance between the point a and the origin, defined as |a| = a if a \geq 0, and -a if a < 0.

Vertical Asymptote

A vertical line x = a where the function f(x) approaches positive or negative infinity.

Horizontal Asymptote

A horizontal line y = b where the function f(x) approaches the value b as x approaches infinity.

Zeros of a Function

The values of x for which f(x) = 0.

Factor Theorem

If p is a zero of f(x), then (x - p) is a factor of f(x).

Quadratic Formula

The solutions of the quadratic equation ax^2 + bx + c = 0 are given by x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Venn Diagram

A diagram that shows all possible logical relations between a finite collection of sets.

Radical Function

A function that contains a root expression, such as \sqrt{x}.

Universal Set (U)

The set that contains all possible objects under consideration in a particular context.

Power Set \mathcal{P}(S)

The set of all possible subsets of a given set S, including the empty set and S itself.

Cardinality |S|

A measure of the 'number of elements' in a set S.

Empty Set \emptyset

The unique set that contains no elements.

Domain of a Function

The set of all possible input values (typically x) for which the function is defined and produces a real output.

Range of a Function

Range of a Function

The set of all possible output values (typically y) that result from using the function's domain.

Interval Notation

A notation for representing an interval as a pair of numbers, using parentheses ( ) for open intervals and brackets [ ] for closed intervals.

Function Composition (f \circ g)(x)

The application of one function to the results of another, defined as f(g(x)).

Inverse Function f^{-1}(x)

A function that reverses the effect of another function, such that if f(x) = y, then f^{-1}(y) = x.

Set Difference (A \setminus B)

The set of elements that are in A but not in B.

Symmetric Difference (A \Delta B)

The set of elements which are in either of the sets A or B, but not in their intersection.

Discriminant (D)

The value derived from the coefficients of a quadratic equation, given by D = b^2 - 4ac, which determines the type of roots the equation has.

Injective Function (One-to-One)

A function where each element of the codomain is mapped to by at most one element of the domain.

Surjective Function (Onto)

A function where every element in the codomain is the image of at least one element in the domain.

Bijective Function

A function that is both injective and surjective, establishing a perfect one-to-one correspondence between the