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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 \([a]\subseteq [b]\).
- Halting Problem
The halting problem is a decision problem in computability...
- Function Terminology
A function is a relation between a set of inputs (called the...
- Halting Problem
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.
3 days 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 \(m>0.\ _\square\)
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4 days ago · In this explainer, we will learn how to find missing lengths in a triangle containing two or three parallel lines using proportionality. Recall that when two parallel lines are cut by a transversal, the resulting corresponding angles are equal.
2 days ago · Definition: The Point of Intersection of Two Lines. The point of intersection of two distinct, nonparallel lines is the single point where they meet or cross. It is the ordered pair of the values of 𝑥 and 𝑦 where the lines meet on the graph and that satisfies the equations of both lines.
4 days ago · A system of inequalities is a set of two or more inequalities in one or more variables. Systems of inequalities are used when a problem requires a range of solutions, and there is more than one constraint on those solutions. Leon is the manager of a textile factory. His workers have a total of 400 man-hours this week for him to allocate.
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