Sets and Functions

The union of two sets A and B, denoted as A ∪ B, is the set of all elements that are in either A or B. Therefore, the union of A = {1, 3, 5} and B = {2, 4, 6} is {1, 2, 3, 4, 5, 6}.

The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are in both A and B. Therefore, the intersection of A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7} is {3, 4, 5}.

A function is a relation that assigns to each element of a set a unique element of another set. In other words, each input value can only have one output value. Therefore, the relation {(1, 2), (2, 3), (3, 4)} is a function, while the other relations are not.

The domain of a function is the set of all possible input values for which the function is defined. Since the expression sqrt(x - 1) is defined only for non-negative values of x - 1, the domain of f(x) = sqrt(x - 1) is [1, ∞).

The range of a function is the set of all possible output values that the function can produce. Since the function f(x) = 2x + 1 is a linear function with no restrictions on the input values, its range is the set of all real numbers, which is [-∞, ∞).

A function is one-to-one if each input value corresponds to a unique output value. The function f(x) = sin(x) is one-to-one because for any two distinct input values x1 and x2, sin(x1) and sin(x2) are distinct.

A function is onto if every element in the range of the function is the output of at least one input value. The function f(x) = cos(x) is onto because for any value y in the range [-1, 1], there exists an input value x such that cos(x) = y.

The inverse function of f(x) is the function that undoes the operation performed by f(x). To find the inverse function of f(x) = 2x + 1, we can solve for x in terms of y: y = 2x + 1. Subtracting 1 from both sides, we get y - 1 = 2x. Dividing both sides by 2, we get x = (y - 1)/2. Therefore, the inverse function of f(x) = 2x + 1 is f^-1(x) = (x - 1)/2.

A composite function is a function that is formed by combining two or more simpler functions. The function f(x) = sin(x^2) is a composite function because it is formed by composing the sine function with the squaring function.

The composition of two functions f(x) and g(x), denoted as f∘g(x), is the function that is obtained by applying g(x) to the input of f(x). Therefore, the composition of f(x) = x^2 and g(x) = x + 1 is h(x) = f∘g(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1.

A bijective function is a function that is both one-to-one and onto. The function f(x) = sin(x) is bijective because it is both one-to-one (each input value corresponds to a unique output value) and onto (every element in the range of the function is the output of at least one input value).

The cardinality of a set is the number of elements in the set. The set A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} contains 10 elements. Therefore, the cardinality of A is 10.

A subset of a set B is a set that contains only elements that are also in B. The set {1, 3, 5} is a subset of B because every element of {1, 3, 5} is also an element of B. The other sets are not subsets of B because they contain elements that are not in B.

The complement of a set A in a universal set U is the set of all elements in U that are not in A. The complement of A = {1, 3, 5} in U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is {2, 4, 6, 8, 10}.

The power set of a set A is the set of all subsets of A. The set {{1, 2}, {3, 4}, {5, 6}} is the power set of the set {1, 2, 3, 4, 5, 6} because it contains all possible subsets of {1, 2, 3, 4, 5, 6}.