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1 day ago · Since the number of ordered non-negative integer triples \((a, b, c)\) satisfying \((1)\) is \(21,\) using the technique of stars and bars, we obtain \[\begin{align} \binom{3+(m-6)-1}{m-6} =\binom{m-4}{m-6}=\binom{m-4}{2}&=21\\ \frac{(m-4)(m-5)}{2!}&=21\\ m^2-9m+20&=42\\ m^2-9m-22&=0\\ (m+2)(m-11)&=0\\ m&=11 \end{align}\] because \(m>0.\ _\square\)
5 days ago · They are disjoint because if \(x \in [a]\) and \(x \in [b],\) then \(x \sim a\) by definition, and by symmetry, \(a \sim x\). Also, \(x \sim b\) by definition. But then \(a \sim b\) by transitivity, so \(a\in [b]\). Every \(y \in [a]\) is also in \([b]\) again by transitivity. This means that \([a]\subseteq [b]\).
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Apr 29, 2024 · In an ordered pair, (𝑥, 𝑦), the first element is related to the second element. Therefore, for a set of ordered pairs (called a relation) to represent a function, we cannot have the same first entry related to two different second entries. We can check this by seeing which relation has distinct first entries.
Definition: Relation. A relation is a rule that associates the elements of one set with the elements of another. Relations can be represented as mapping diagrams, ordered pairs, tables of values, equations, or graphs. Simply put, a relation produces one or more outputs for every input.
May 2, 2024 · In mathematics, a binary relation associates elements of one set, called the domain, with elements of another set, called the codomain. [1] A binary relation over sets and is a set of ordered pairs consisting of elements from and from . [2]
May 4, 2024 · An ordered pair is made up of the two coordinates and is written like this: (x, y). The first number is read in terms of the x-axis and the second number is identified in terms of the y-axis. An example of an ordered pair is (5, 7).
May 1, 2024 · An ordered pair of real numbers are two real numbers in an assigned order. Every point on a coordinate plane can be associated with a unique ordered pair of real numbers, usually ( x, y) , also called coordinates. The first value, x, is commonly called the x-coordinate, and the second value, y is called the y-coordinate.
