A relation R on a set A is reflexive if (a, a) belongs to R for every a in A. It is symmetric if (a, b) in R implies (b, a) in R. It is transitive if (a, b) in R and (b, c) in R together imply (a, c) in R. A relation that is reflexive, symmetric and transitive at the same time is called an equivalence relation.

Reflexive: for any integer a, a - a = 0, which is divisible by 3, so (a, a) in R. Symmetric: if (a - b) is divisible by 3, then (b - a) = -(a - b) is also divisible by 3, so aRb implies bRa. Transitive: if (a - b) and (b - c) are both divisible by 3, then their sum (a - b) + (b - c) = (a - c) is also divisible by 3, so aRb and bRc imply aRc. Since R is reflexive, symmetric and transitive, it is an equivalence relation.

A function f from A to B is one-one (injective) if distinct elements of A have distinct images, that is, f(x1) = f(x2) implies x1 = x2. It is onto (surjective) if every element of B has at least one pre-image in A, that is, the range of f equals the codomain B. A function that is both one-one and onto is called bijective (a one-to-one correspondence).

One-one: suppose f(x1) = f(x2). Then 2x1 + 3 = 2x2 + 3, so 2x1 = 2x2 and x1 = x2; hence f is one-one. Onto: let y be any real number in the codomain. Solving y = 2x + 3 gives x = (y - 3)/2, which is a real number, and f((y - 3)/2) = y. So every y has a pre-image, and f is onto. Being both one-one and onto, f is a bijection.

(g o f)(x) means g(f(x)). So (g o f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2. (f o g)(x) means f(g(x)). So (f o g)(x) = f(g(x)) = f(2x) = 2x + 1. Note that (g o f)(x) is not equal to (f o g)(x), so composition of functions is not commutative in general.

(g o f)(x) = g(f(x))

Since f is a bijection, its inverse exists. Let y = f(x) = 2x + 3. Solve for x in terms of y: y - 3 = 2x, so x = (y - 3)/2. Therefore the inverse function is f inverse (y) = (y - 3)/2, or writing in terms of x, f inverse (x) = (x - 3)/2. We can check that f(f inverse (x)) = x.

f inverse (x) = (x - 3)/2

The principal value branch of sin inverse is [-pi/2, pi/2]. Since sin(pi/6) = 1/2, the principal value of sin inverse (1/2) = pi/6. The principal value branch of cos inverse is [0, pi]. Since cos(2pi/3) = -1/2, the principal value of cos inverse (-1/2) = 2pi/3.

The principal values are: tan inverse (1) = pi/4, cos inverse (1/2) = pi/3, and sin inverse (1/2) = pi/6. Adding them: pi/4 + pi/3 + pi/6. Taking the LCD 12: 3pi/12 + 4pi/12 + 2pi/12 = 9pi/12 = 3pi/4.

Let sin inverse (x) = theta, so sin theta = x, where theta lies in [-pi/2, pi/2]. Then x = sin theta = cos(pi/2 - theta). Since (pi/2 - theta) lies in [0, pi], we have cos inverse (x) = pi/2 - theta = pi/2 - sin inverse (x). Rearranging gives sin inverse (x) + cos inverse (x) = pi/2.

sin inverse (x) + cos inverse (x) = pi/2

First combine tan inverse (2) + tan inverse (3). Using tan inverse (x) + tan inverse (y) = pi + tan inverse ((x + y)/(1 - xy)) when xy > 1: here x = 2, y = 3, xy = 6 > 1, so tan inverse (2) + tan inverse (3) = pi + tan inverse ((5)/(1 - 6)) = pi + tan inverse (-1) = pi - pi/4 = 3pi/4. Adding tan inverse (1) = pi/4: total = pi/4 + 3pi/4 = pi. Hence proved.

tan inverse x + tan inverse y = tan inverse((x + y)/(1 - xy))

The principal value branch of cos inverse is [0, pi], but 7pi/6 does not lie in this interval. First find cos(7pi/6) = cos(pi + pi/6) = -cos(pi/6) = -sqrt(3)/2. Now cos inverse (-sqrt(3)/2) = 5pi/6, since 5pi/6 lies in [0, pi] and cos(5pi/6) = -sqrt(3)/2. Therefore cos inverse (cos(7pi/6)) = 5pi/6.

For sin inverse (x): the domain is [-1, 1] and the principal value range is [-pi/2, pi/2]. For tan inverse (x): the domain is all real numbers R and the principal value range is the open interval (-pi/2, pi/2). Inverse trigonometric functions are defined by restricting the domain of the corresponding trigonometric functions so that they become one-one.

A diagonal matrix is a square matrix in which all the non-diagonal (off-diagonal) elements are zero. A scalar matrix is a diagonal matrix in which all the diagonal elements are equal (to the same non-zero scalar). An identity (unit) matrix is a scalar matrix in which all the diagonal elements are equal to 1; it is denoted by I and acts as the multiplicative identity, since AI = IA = A.

Two matrices are equal when their corresponding elements are equal. Comparing: x + y = 6 and x - y = 2. Adding these two equations: 2x = 8, so x = 4. Subtracting the second from the first: 2y = 4, so y = 2. Hence x = 4 and y = 2.

To multiply, each element of a row of A is multiplied by the corresponding element of a column of B and summed. AB = [[(1)(2) + (2)(1), (1)(0) + (2)(3)], [(3)(2) + (4)(1), (3)(0) + (4)(3)]] = [[2 + 2, 0 + 6], [6 + 4, 0 + 12]] = [[4, 6], [10, 12]].

(AB)ij = sum of (row i of A)(column j of B)

The transpose of a matrix A, denoted A' (or A^T), is the matrix obtained by interchanging its rows and columns. Two properties are: (1) (A')' = A, that is, the transpose of the transpose is the original matrix; and (2) (AB)' = B' A', the transpose of a product equals the product of the transposes in reverse order. Also (A + B)' = A' + B' and (kA)' = k A'.

(AB)' = B' A'

A square matrix A is symmetric if A' = A, that is, aij = aji for all i and j (it is unchanged on taking its transpose). A square matrix A is skew-symmetric if A' = -A, that is, aij = -aji for all i and j; this forces all its diagonal elements to be zero. Every square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

symmetric: A' = A; skew-symmetric: A' = -A

Write A = (1/2)(A + A') + (1/2)(A - A'). Let P = (1/2)(A + A') and Q = (1/2)(A - A'). Then P' = (1/2)(A + A')' = (1/2)(A' + A) = P, so P is symmetric. And Q' = (1/2)(A - A')' = (1/2)(A' - A) = -(1/2)(A - A') = -Q, so Q is skew-symmetric. Since A = P + Q, every square matrix is the sum of a symmetric matrix P and a skew-symmetric matrix Q.

A = (1/2)(A + A') + (1/2)(A - A')

If a matrix has 12 elements, then the order (number of rows x number of columns) must be a pair of factors whose product is 12. The possible orders are: 1 x 12, 12 x 1, 2 x 6, 6 x 2, 3 x 4 and 4 x 3. So there are 6 possible orders. (In general, the number of possible orders equals the number of ways of writing the total number of elements as a product of two natural numbers.)

First find 2A = [[2, 4],[6, 8]]. Then subtract B element by element: 2A - B = [[2 - 2, 4 - 1],[6 - 0, 8 - 5]] = [[0, 3],[6, 3]].