Relations and Functions in Mathematics

A function where each element in the domain maps to a unique element in the codomain.

A function where every element in the codomain has at least one corresponding element in the domain.

A function where different inputs map to different outputs.

A function where every element in the codomain has at least one pre-image in the domain

A function that maps a real number to the greatest integer less than or equal to that number.

A relation R in a set A is reflexive if (a, a) ∈ R for every element a in A.

A relation R in a set A is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.

A relation R in a set A is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

The relation of integers where the subtraction result of x from y is always an integer; therefore always reflexive, symmetric, and transitive

A relation R on a set A is reflexive if (a, a) ∈ R for every element a ∈ A.

A relation R on a set A is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b ∈ A.

A relation R on a set A is transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R for all a, b, c ∈ A.

A relation that is reflexive, symmetric, and transitive.

In a relation R between elements a and b, (a, b) ∈ R signifies that element a is related to element b under relation R.

A set of ordered pairs from a set to itself.

{(1,2), (2,1)}

Books with same number of pages.

Elements with even difference between them in a set is related

Reflexive, symmetric, transitive, equivalence is classifications of relations based on their properties.

Points at equal distances from a fixed point (origin) in the plane form an equivalence relation.

The angle in the interval [-π/2, π/2] whose sine is -1/2.

The angle in the interval [0, π] whose cosine is √3/2.

The angle in the interval [-π/2, π/2] whose cosecant is 2.

The angle in the interval (-π/2, π/2) whose tangent is -√3.

The angle in the interval [0, π] whose cosine is -1/2.

The function that maps any real number to its absolute value. It is defined as f(x) = |x|, where |x| is x if x is positive or 0, and |x| is -x if x is negative.

The function that returns the sign of a real number. It is defined as f(x) = 1 for x > 0, f(x) = 0 for x = 0, and f(x) = -1 for x < 0.

A function that is both one-to-one and onto. This means each element in the domain maps to a unique element in the codomain, and every element in the codomain has a corresponding element in the domain.

Functions that return the angle (in radians) corresponding to a given trigonometric value (sine, cosine, tangent, etc.). For example, arcsin(x) returns the angle whose sine is x.

The angle whose secant is 2/√3. Since secant is the reciprocal of cosine, find the angle whose cosine is √3/2. This angle is π/6 radians.

The angle whose cotangent is √3. Since cotangent is the reciprocal of tangent, find the angle whose tangent is 1/√3. This angle is π/6 radians.

The range of y is 0 ≤ y ≤ π. This is because the sine function has a range of [-1, 1], and the inverse sine function only returns angles within the principal branch of the sine function, from -π/2 to π/2.

tan⁻¹(√3) = π/3 and sec⁻¹(-2) = -π/3. So, the answer is π/3 - (-π/3) = 2π/3.

Expressing an inverse trigonometric function in terms of its base trigonometric function. For example, expressing $\cot^{-1} \sqrt{x^2-1}$ as $\sec^{-1} x$.

Show that two trigonometric expressions involving inverse functions are equivalent for a specific range of values. This involves manipulation of trigonometric identities and properties of inverse functions.

Simplify a given expression containing inverse trigonometric functions by using trigonometric identities, algebraic manipulation, and properties of inverse functions.

Calculate the value of an expression involving multiple inverse and regular trigonometric functions, often including nested functions. This involves applying the properties and definitions of inverse functions sequentially.

The dimensions of a matrix, represented as rows x columns.

The individual values within a matrix. They can be numbers, variables, or expressions.

Using 'aij' to represent the element in the i-th row and j-th column.

Creating a matrix using a formula that defines the relationship between row and column indices to determine each element's value.

A system of equations represented using matrices, where the unknown variables are arranged into a matrix.

The inverse sine function, denoted as sin⁻¹(x) or arcsin(x), finds the angle whose sine is x. It's the inverse operation of the sine function, giving an angle within the range of -π/2 to π/2.

The inverse tangent function, written as tan⁻¹(x) or arctan(x), finds the angle whose tangent is x. It's the inverse of the tangent function, resulting in an angle within the range of -π/2 to π/2.

The inverse cosine function, represented as cos⁻¹(x) or arccos(x), finds the angle whose cosine is x. It's the inverse of the cosine function, yielding an angle in the range of 0 to π.

Within the inverse trigonometric functions, the principal branch refers to the specific range of output angles that the function provides. For example, the principal branch of arcsine is from -π/2 to π/2.

To simplify expressions involving inverse trigonometric functions, use the inverse operation to get the original angle, check if the angle lies within the principal branch, and find an equivalent angle that's within the principal branch if necessary. This ensures the simplified expression is consistent with the function's definition.

To check whether a relation R on a set A is reflexive, symmetric, or transitive, you need to examine the ordered pairs in R and see if they satisfy the conditions for each property.