inverse function L’Hospital’s Rule
If f is an injective (i.e., one-to-one) function, then f is said to be invertible, and its inverse is the function g satisfying g(f(x)) = f(g(x)) = x for every x in the domain of f.
Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y.
Given a conditional statement, i.e., a statement of the form “if A then B,” or “A implies B,” its inverse is “not A implies not B.” A conditional neither implies nor is implied by its inverse. However, the converse of a conditional and its inverse are logically equivalent, since they are contrapositives of each other.
A real number which is not a rational number, i.e., which cannot be expressed as a ratio of integers. Examples: p (the ratio of the circumference of a circle to its diameter) and the square root of 2 (ratio of the length of the diagonal of a square to the length of one side). The irrational numbers are uncountably infinite.
See reflexive relation.
A morphism that is both injective and surjective, that is, both “one-to-one” and “onto.” An isomorphism from a structure to itself is called an automorphism.
A function that is order preserving and increasing.
A binary operation whose value on two elements a and b of a lattice is the least upper bound of a and b, denoted a b.
An element a of a lattice is called join irreducible if whenever a = b c then a = b or a = c. More generally, a is called strictly join irreducible if whenever a is the join of a subset X of the lattice, then a is an element of X.
Cf. join prime, meet irreducible, meet prime.
A function f on a lattice L is called a join-morphism if for every a and b in L we have f(a b) = f(a) f(b). If f has an inverse that is also a join-morphism, then f is called a join-isomorphism. A join-isomorphism from a lattice to itself is called a join-automorphism.
Cf. join, meet-morphism.
An element a of a lattice is called join prime if whenever a b c then a b or a c. Also, a is called strictly join prime if
Cf. join irreducible, meet prime, meet irreducible.
The laws of planetary motion discovered by Johannes Kepler. See the article for an exposition.
If a is an infinite cardinal and the cofinality of a is not greater than b, then ab is strictly greater than a.
Many Latin terms and phrases are used in mathematical writing. See the article for a full exposition.
A partially ordered set in which each pair of elements has a least upper bound, called the join, and a greatest lower bound, called the meet. The join of two elements x and y is denoted by x y, and the meet by x y.
A lattice may be equivalently defined as a set with an algebra having two binary operations (meet) and (join) defined on it, satisfying
A lattice is called complete if every subset of the lattice has a least upper bound and a greatest lower bound.
- x x = x = x x, and
- x (x y) = x = x (x y)
laws of exponents
The following rules govern the behavior of exponents:
Additionally, x0 = 1 for all x except 0, and 00 = 0 or is left undefined (i.e., it is an indeterminate form).
Cf. rational exponent.
laws of logarithms
The following rules governing the behavior of logarithms are easily derived, and very useful in calculations:
Students often confuse these rules, so it is worth memorizing them as “the sum of the logs is the log of the product – and not the product of the logs,” etc.
- log A + log B = log (A×B)
- log A – log B = log (A/B)
- log Ap = p × log A
least upper bound
An upper bound which is less than or equal to every other upper bound.
Cf. greatest lower bound.
least upper bound axiom
“Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.
On a right triangle, the sides adjacent to the right angle are called the legs.
If a limit is in one of the indeterminate forms “infinity over infinity” or ”zero over zero,” then we have
That is, the limit is the same after taking the derivatives of both the numerator and denominator. L’Hospital’s Rule may be applied as many times as needed. Students often misapply the Rule by using it when the limit is not in one of the above indeterminate forms. This often results in an incorrect evaluation of the limit.