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In the context of proofs, this phrase is often seen in induction arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence. necessary and sufficient
A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor. [5] In order to find the greatest common divisor, the Euclidean algorithm or prime factorization can be used. The Euclidean algorithm is ...
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2.
For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗, we have f(x ∗) > f(x), and x ∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
Example 3: In the fence < > < > < > …, all the are minimal and all are maximal, as shown in the image. Example 4: Let A be a set with at least two elements and let S = { { a } : a ∈ A } {\displaystyle S=\{\{a\}~:~a\in A\}} be the subset of the power set ℘ ( A ) {\displaystyle \wp (A)} consisting of singleton subsets , partially ordered by ...
Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. [1] The supremum (abbreviated sup ; pl. : suprema ) of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the least element in P {\displaystyle P} that is greater than or equal to each element of S , {\displaystyle S,} if such an ...
In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions.
def – define or definition. deg – degree of a polynomial, or other recursively-defined objects such as well-formed formulas. (Also written as ∂.) del – del, a differential operator. (Also written as.) det – determinant of a matrix or linear transformation. DFT – discrete Fourier transform.