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Critics point out that priors are subjective starting points influenced by background knowledge. The definition of the source population matters significantly: whether Steve is drawn randomly from national census data or is someone personally known to the observer changes the baseline widths in probability diagrams.

Conditions: Application of base rates to specific individual profiles; Ambiguity exists regarding the sampling frame (random vs. personal acquaintance)

The entropy method derives weights directly from the data pattern itself, measuring the dispersion of alternatives. Unlike AHP, which requires subjective pairwise comparison judgments from decision makers, the entropy method is objective and data-driven.

Conditions: Comparing the entropy method with the Analytic Hierarchy Process (AHP).; The decision matrix contains the necessary numerical data.

Optimal substructure means that the optimal solution to the overall problem can be constructed by efficiently combining the optimal solutions of its subproblems. In the route example, once the shortest SF→LA and LA→SD segments are known, they combine to give the shortest SF→SD route.

Conditions: The problem can be divided into subproblems.; The optimal solution of the main problem depends on the optimal solutions of its subproblems.

To translate the graph of $y=f(x)$ four units left and three units down, replace $x$ with $x+4$ inside the function argument and subtract 3 outside the function. The final transformed equation is $y=f(x+4)-3$.

Conditions: The starting graph has equation $y=f(x)$.; The intended transformation is a horizontal translation 4 units left and a vertical translation 3 units down.

The weight loss rate in the close and damp condition is approximately $1/5$ of the weight loss rate in the permanently soaked in water and wet and dry alternately conditions.

Conditions: The ratio refers to the weight loss rate metric.; The comparison is made against both permanently soaked in water and wet-and-dry-alternately environments.

To find the coordinates of a vector in an alternative basis, express the vector as a linear combination of the new basis vectors. The scalars in this combination are the coordinates. This method requires that the alternative basis vectors are linearly independent and the target vector lies in their span. For example, expressing $(3,2)$ in the basis $b_1=(2,1)$ and $b_2=(-1,1)$ involves solving $(5/3)b_1+(1/3)b_2=(3,2)$, yielding coordinates $(5/3,1/3)$. Note that simple projection only works for orthogonal bases.

Conditions: The alternative basis vectors are linearly independent.; The target vector lies in the span of the alternative basis.

When using u-substitution in a definite integral, the original limits of integration in terms of $x$ must be converted into new limits in terms of $u$. This is done by substituting the original $x$-bounds into the substitution equation $u(x)$.

Conditions: The integral is a definite integral.; A substitution $u = u(x)$ is being used.; The original limits of integration are given in terms of $x$.

To resolve the $0/0$ indeterminate form of $\lim_{x\to5}\frac{x-5}{\sqrt{x+4}-3}$, multiply the numerator and denominator by the conjugate $\sqrt{x+4}+3$. This application of the difference of squares simplifies the denominator to $x-5$, allowing cancellation with the numerator. The resulting expression $\sqrt{x+4}+3$ evaluates to $6$ as $x$ approaches $5$.

Conditions: The limit is as $x \to 5$.; The expression involves a radical in the denominator causing a $0/0$ form.; The conjugate is $\sqrt{x+4}+3$.

Eigenvectors are special because they do not rotate off their original line (span) during a transformation; they only stretch, shrink, or flip. The zero vector is excluded from this definition because it maps to itself under any linear transformation ($A\vec{0}=\vec{0}$), providing no directional information about the transformation's effect.

Conditions: Vector $\vec{v}$ must be non-zero ($\vec{v} \neq \vec{0}$); Transformation is linear

The example proves no matching is possible because the highlighted subset of three girls has a neighborhood of only two boys. Since |S| = $3 > 2$ = |$E(S)$|, this subset is a bottleneck, and by the Bottleneck Lemma, no matching can cover all girls.

Conditions: A bipartite graph with left set G and right set B.; A highlighted subset S of three girls.; Their collective neighborhood $E(S)$ contains only two boys.

The vertical line test is a graphical method used to determine whether a curve represents a function of $x$. A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Conditions: The graph is plotted on a standard Cartesian coordinate system.; The independent variable is $x$.

Rolle's theorem is a specific instance of the mean value theorem where the function values at the endpoints are equal ($f(a) = f(b)$). Under the standard conditions of continuity on $[a,b]$ and differentiability on $(a,b)$, this equality forces the average rate of change to zero. Consequently, the mean value theorem guarantees the existence of a point $c \in (a,b)$ such that $f'(c) = 0$, representing a horizontal tangent.

Conditions: $f$ is continuous on $[a,b]$; $f$ is differentiable on $(a,b)$; $f(a) = f(b)$