UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
The intersection of two closed intervals is another closed interval.
If two intervals are disjoint, then there is a coordinate in which they are disjoint.
The union of all boxes of the form $[-n, n]^n$ is the entire space.
The image of a box under an affine map is a box.
The image of a box under multiplication by a scalar is a box.
If $f$ is a continuous function from the rectangle $[a,b] \times [c,d]$ to $\mathbb{R}$, then the function $g$ defined by $g(x,y) = f(y,x)$ is continuous on the rectangle $[c,d] \times [a,b]$.
A set $S$ is an interval if and only if for all $a, b \in S$ and all $x$ such that $a \leq x \leq b$, we have $x \in S$.
The intersection of two intervals is an interval.
The closed box $[a,b]$ and the open box $(a,b)$ are intervals.
The empty set is an interval.
The set of all real numbers is an interval.
If $s$ is an interval, $a, b \in s$, and $x$ is between $a$ and $b$ in each coordinate, then $x \<in> s$.
If $S$ is an interval in $\mathbb{R}^n$, and $x, y_1, \ldots, y_n \in S$, then the point $(y_1, \ldots, y_n)$ is in $S$.
If $S$ is an interval and $x$ is in the projection of $S$ onto each coordinate axis, then $x \<in> S$.
The unit cube $[0,1]^n$ is nonempty.
The unit box $[0,1]^n$ is nonempty.
The empty set is the same as the interval $[1,0]$.
If $S$ is an interval, then $[a,b] \subseteq S$ if and only if $[a,b] = \emptyset$ or $a,b \in S$.
If $X$ and $Y$ are intervals and $X \cap Y \neq \emptyset$, then $X \cup Y$ is an interval.
If $X$ is an interval, then $X + x$ is an interval.
If $X$ is an interval, then $-X$ is an interval.
The image of an interval under the map $x \mapsto -x$ is an interval.
If $X$ is an interval, then so is $-x + X$.
The image of an interval under a translation is an interval.
The image of an interval under the translation map $x \mapsto x - a$ is an interval.
The image of an interval under a translation is an interval.
The closed ball is an interval.
The open ball in $\mathbb{R}$ is an interval.
If $s$ is a bounded set in $\mathbb{R}^n$, then the set of inner products $\{x \cdot a \mid x \in s\}$ is bounded above.
If $s$ is a bounded set in $\mathbb{R}^n$, then the set of inner products $\{x \cdot a \mid x \in s\}$ is bounded below.
If $f$ is a continuous function from a topological space $F$ to a normed vector space $E$, then the function $x \mapsto \\|f(x)\\|$ is continuous.
If $f$ and $g$ are continuous functions from a topological space $X$ to a normed vector space $V$, then the function $x \mapsto \langle f(x), g(x) \rangle$ is continuous.
If $f$ is continuous on $S$, then the function $x \mapsto \\|f(x)\\|$ is continuous on $S$.
If $f$ and $g$ are continuous functions from a topological space $X$ to a real inner product space $Y$, then the function $x \mapsto \langle f(x), g(x) \rangle$ is continuous.
The set of points $x$ such that $a \cdot x < b$ is open.
The set of points $x$ such that $a \cdot x > b$ is open.
The set of points in $\mathbb{R}^n$ whose $i$th coordinate is less than $a$ is open.
The set of points in $\mathbb{R}^n$ whose $i$th coordinate is greater than $a$ is open.
The set of points in $\mathbb{R}^n$ that are less than a given point $a$ is the intersection of the half-spaces $\{x \in \mathbb{R}^n \mid x_i < a_i\}$ for $i = 1, \ldots, n$.
The sets of points less than and greater than a given point are open.
The inner product of a vector $a$ with a vector $x$ is a continuous function of $x$.
The set of points $x$ such that $a \cdot x \leq b$ is closed.
The set of points $x$ such that $a \cdot x \geq b$ is closed.
The set of points $x$ such that $a \cdot x = b$ is closed.
The set of points in $\mathbb{R}^n$ whose $i$th coordinate is less than or equal to $a$ is closed.
The set of points in $\mathbb{R}^n$ whose $i$th coordinate is greater than or equal to $a$ is closed.
The set of points in $\mathbb{R}^n$ whose coordinates are less than or equal to the coordinates of a given point $b$ is closed.
The set of points in $\mathbb{R}^n$ whose coordinates are all greater than or equal to the coordinates of a given point $a$ is closed.
If $a \neq 0$, then the interior of the halfspace $\{x \mid a \cdot x \leq b\}$ is the halfspace $\{x \mid a \cdot x < b\}$.
The interior of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$ is the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$.
If $a \neq 0$, then the closure of the half-space $\{x \in \mathbb{R}^n \mid a \cdot x < b\}$ is the half-space $\{x \in \mathbb{R}^n \mid a \cdot x \leq b\}$.
If $a \neq 0$, then the closure of the half-space $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$ is the half-space $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$.
If $a \neq 0$, then the interior of the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$ is empty.
If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \mid a \cdot x \leq b\}$ is the hyperplane $\{x \mid a \cdot x = b\}$.
If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$ is the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$.
If $a$ or $b$ is nonzero, then the frontier of the halfspace $\{x \mid a \cdot x < b\}$ is the hyperplane $\{x \mid a \cdot x = b\}$.
If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$ is the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$.
If $S$ is a connected set and $x, y \in S$ satisfy $a \cdot x \leq b \leq a \cdot y$, then there exists $z \in S$ such that $a \cdot z = b$.
If $S$ is a connected set in $\mathbb{R}^n$ and $x, y \in S$ with $x_k \leq a \leq y_k$, then there exists $z \in S$ such that $z_k = a$.
If $f$ converges to $l$ and $f(x)_i \leq b$ for all $x$ in some neighborhood of $l$, then $l_i \leq b$.
If $f$ converges to $l$ and $b \leq f(x)_i$ for all $x$ in some neighborhood of $l$, then $b \<leq l_i$.
If $f$ is a function from a topological space to a Euclidean space, and if $f$ converges to $l$ and $f(x)_i = b$ for all $x$ in some neighborhood of $l$, then $l_i = b$.
The box $[a,b]$ is open.
The closed interval $[a, b]$ is closed.
The interior of a closed box is the open box.
The closed box $[a,b]$ is bounded.
The box $[a,b]$ is bounded.
The closed and open unit cubes in $\mathbb{R}^n$ are not the entire space.
The unit interval is not the whole space.
If the box $[a,b]$ is nonempty, then its midpoint $(a+b)/2$ is in the box.
If $x$ and $y$ are in the closed box $[a,b]$ and $0 < e \leq 1$, then $e x + (1-e) y$ is in the open box $(a,b)$.
The closure of a closed interval is the closed interval.
The closure of a non-empty box is the box itself.
If $S$ is a bounded subset of $\mathbb{R}^n$, then there exists a vector $a \in \mathbb{R}^n$ such that $S$ is contained in the box $[-a, a]$.
If $S$ is a bounded subset of $\mathbb{R}^n$, then there exists a vector $a \in \mathbb{R}^n$ such that $S \subseteq [-a, a]$.
The frontier of a closed box is the boundary of the box.
The frontier of a box is the boundary of the box.
If $c$ and $d$ are two points in $\mathbb{R}^n$ such that the box $[c,d]$ is nonempty, then the intersection of the box $[a,b]$ with the box $[c,d]$ is empty if and only if the intersection of the box $[a,b]$ with the box $[c,d]$ is empty.
If the range of a sequence of vectors is bounded, then for every subset $d$ of the basis of the vector space, there exists a vector $l$ and a strictly increasing sequence of natural numbers $r$ such that for every $\epsilon > 0$, there exists $n$ such that for all $i \in d$, we have $\|f(r(n))_i - l_i\| < \epsilon$.
The closed box $[a,b]$ is compact.
A set $S$ is an interval and compact if and only if there exist $a$ and $b$ such that $S = [a, b]$.
For any two points $x$ and $y$ in $\mathbb{R}^n$, the distance between the $i$th coordinates of $x$ and $y$ is less than or equal to the distance between $x$ and $y$.
If $S$ is an open set in $\mathbb{R}$, then the set $\{x \in \mathbb{R}^n \mid x_i \in S\}$ is open in $\mathbb{R}^n$.
A sequence of vectors $f_n$ converges to a vector $l$ if and only if each of its components converges to the corresponding component of $l$.
A function $f: \mathbb{R}^n \to \mathbb{R}^n$ is continuous if and only if each of its component functions is continuous.
A function $f$ is continuous on a set $S$ if and only if each of its components is continuous on $S$.
A function $f'$ is linear if and only if each of its components is linear.
A linear map $f'$ is bounded if and only if each of its components is bounded.
If $a$ and $b$ are vectors such that $a_i \leq b_i$ for all $i$, then the clamp function maps $x$ to a point in the box $[a, b]$.
If $x$ is in the closed box $[a,b]$, then clamping $x$ to $[a,b]$ does nothing.
If $a$ and $b$ are two points in $\mathbb{R}^n$ such that $a_i > b_i$ for some $i$, then the clamp function returns $a$ for all points.
The distance between two clamped points is less than or equal to the distance between the original points.
If $f$ is continuous on the closed box $[a, b]$, then the function $x \mapsto f(clamp(a, b, x))$ is continuous at $x$.
Suppose $f$ is a continuous function defined on the closed box $[a,b]$. Then the function $g$ defined by $g(x) = f(clamp(a,b,x))$ is continuous on any set $S$.
If $f$ is a function from a closed box to a metric space, then the function that maps each point in the box to the closest point in the image of $f$ is bounded.
If $f$ is a continuous function on a closed box $[a,b]$, then $f$ is equal to its extension by continuity.
If $f$ is continuous on the closed interval $[a,b]$, then the extension of $f$ to the whole real line is continuous on any set $S$.
There exists a sequence of open sets $B_n$ such that $B_n$ is injective and for any open set $S$, there exists a finite set $k$ such that $S = \bigcup_{n \in k} B_n$.
Every separable metric space is the closure of a countable subset.
The diameter of a closed ball is $0$ if the radius is negative, and $2r$ if the radius is nonnegative.

The intersection of two closed intervals is another closed interval.

If two intervals are disjoint, then there is a coordinate in which they are disjoint.

The union of all boxes of the form $[-n, n]^n$ is the entire space.

The image of a box under an affine map is a box.

The image of a box under multiplication by a scalar is a box.

If $f$ is a continuous function from the rectangle $[a,b] \times [c,d]$ to $\mathbb{R}$, then the function $g$ defined by $g(x,y) = f(y,x)$ is continuous on the rectangle $[c,d] \times [a,b]$.

A set $S$ is an interval if and only if for all $a, b \in S$ and all $x$ such that $a \leq x \leq b$, we have $x \in S$.

The intersection of two intervals is an interval.

The closed box $[a,b]$ and the open box $(a,b)$ are intervals.

The empty set is an interval.

The set of all real numbers is an interval.

If $s$ is an interval, $a, b \in s$, and $x$ is between $a$ and $b$ in each coordinate, then $x \<in> s$.

If $S$ is an interval in $\mathbb{R}^n$, and $x, y_1, \ldots, y_n \in S$, then the point $(y_1, \ldots, y_n)$ is in $S$.

If $S$ is an interval and $x$ is in the projection of $S$ onto each coordinate axis, then $x \<in> S$.

The unit cube $[0,1]^n$ is nonempty.

The unit box $[0,1]^n$ is nonempty.

The empty set is the same as the interval $[1,0]$.

If $S$ is an interval, then $[a,b] \subseteq S$ if and only if $[a,b] = \emptyset$ or $a,b \in S$.

If $X$ and $Y$ are intervals and $X \cap Y \neq \emptyset$, then $X \cup Y$ is an interval.

If $X$ is an interval, then $X + x$ is an interval.

If $X$ is an interval, then $-X$ is an interval.

The image of an interval under the map $x \mapsto -x$ is an interval.

If $X$ is an interval, then so is $-x + X$.

The image of an interval under a translation is an interval.

The image of an interval under the translation map $x \mapsto x - a$ is an interval.

The image of an interval under a translation is an interval.

The closed ball is an interval.

The open ball in $\mathbb{R}$ is an interval.

If $s$ is a bounded set in $\mathbb{R}^n$, then the set of inner products $\{x \cdot a \mid x \in s\}$ is bounded above.

If $s$ is a bounded set in $\mathbb{R}^n$, then the set of inner products $\{x \cdot a \mid x \in s\}$ is bounded below.

If $f$ is a continuous function from a topological space $F$ to a normed vector space $E$, then the function $x \mapsto \|f(x)\|$ is continuous.

If $f$ and $g$ are continuous functions from a topological space $X$ to a normed vector space $V$, then the function $x \mapsto \langle f(x), g(x) \rangle$ is continuous.

If $f$ is continuous on $S$, then the function $x \mapsto \|f(x)\|$ is continuous on $S$.

If $f$ and $g$ are continuous functions from a topological space $X$ to a real inner product space $Y$, then the function $x \mapsto \langle f(x), g(x) \rangle$ is continuous.

The set of points $x$ such that $a \cdot x < b$ is open.

The set of points $x$ such that $a \cdot x > b$ is open.

The set of points in $\mathbb{R}^n$ whose $i$th coordinate is less than $a$ is open.

The set of points in $\mathbb{R}^n$ whose $i$th coordinate is greater than $a$ is open.

The set of points in $\mathbb{R}^n$ that are less than a given point $a$ is the intersection of the half-spaces $\{x \in \mathbb{R}^n \mid x_i < a_i\}$ for $i = 1, \ldots, n$.

The sets of points less than and greater than a given point are open.

The inner product of a vector $a$ with a vector $x$ is a continuous function of $x$.

The set of points $x$ such that $a \cdot x \leq b$ is closed.

The set of points $x$ such that $a \cdot x \geq b$ is closed.

The set of points $x$ such that $a \cdot x = b$ is closed.

The set of points in $\mathbb{R}^n$ whose $i$th coordinate is less than or equal to $a$ is closed.

The set of points in $\mathbb{R}^n$ whose $i$th coordinate is greater than or equal to $a$ is closed.

The set of points in $\mathbb{R}^n$ whose coordinates are less than or equal to the coordinates of a given point $b$ is closed.

The set of points in $\mathbb{R}^n$ whose coordinates are all greater than or equal to the coordinates of a given point $a$ is closed.

If $a \neq 0$, then the interior of the halfspace $\{x \mid a \cdot x \leq b\}$ is the halfspace $\{x \mid a \cdot x < b\}$.

The interior of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$ is the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$.

If $a \neq 0$, then the closure of the half-space $\{x \in \mathbb{R}^n \mid a \cdot x < b\}$ is the half-space $\{x \in \mathbb{R}^n \mid a \cdot x \leq b\}$.

If $a \neq 0$, then the closure of the half-space $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$ is the half-space $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$.

If $a \neq 0$, then the interior of the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$ is empty.

If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \mid a \cdot x \leq b\}$ is the hyperplane $\{x \mid a \cdot x = b\}$.

If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x \geq b\}$ is the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$.

If $a$ or $b$ is nonzero, then the frontier of the halfspace $\{x \mid a \cdot x < b\}$ is the hyperplane $\{x \mid a \cdot x = b\}$.

If $a \neq 0$ or $b \neq 0$, then the frontier of the halfspace $\{x \in \mathbb{R}^n \mid a \cdot x > b\}$ is the hyperplane $\{x \in \mathbb{R}^n \mid a \cdot x = b\}$.

If $S$ is a connected set and $x, y \in S$ satisfy $a \cdot x \leq b \leq a \cdot y$, then there exists $z \in S$ such that $a \cdot z = b$.

If $S$ is a connected set in $\mathbb{R}^n$ and $x, y \in S$ with $x_k \leq a \leq y_k$, then there exists $z \in S$ such that $z_k = a$.

If $f$ converges to $l$ and $f(x)_i \leq b$ for all $x$ in some neighborhood of $l$, then $l_i \leq b$.

If $f$ converges to $l$ and $b \leq f(x)_i$ for all $x$ in some neighborhood of $l$, then $b \<leq l_i$.

If $f$ is a function from a topological space to a Euclidean space, and if $f$ converges to $l$ and $f(x)_i = b$ for all $x$ in some neighborhood of $l$, then $l_i = b$.

The box $[a,b]$ is open.

The closed interval $[a, b]$ is closed.

The interior of a closed box is the open box.

The closed box $[a,b]$ is bounded.

The box $[a,b]$ is bounded.

The closed and open unit cubes in $\mathbb{R}^n$ are not the entire space.

The unit interval is not the whole space.

If the box $[a,b]$ is nonempty, then its midpoint $(a+b)/2$ is in the box.

If $x$ and $y$ are in the closed box $[a,b]$ and $0 < e \leq 1$, then $e x + (1-e) y$ is in the open box $(a,b)$.

The closure of a closed interval is the closed interval.

The closure of a non-empty box is the box itself.

If $S$ is a bounded subset of $\mathbb{R}^n$, then there exists a vector $a \in \mathbb{R}^n$ such that $S$ is contained in the box $[-a, a]$.

If $S$ is a bounded subset of $\mathbb{R}^n$, then there exists a vector $a \in \mathbb{R}^n$ such that $S \subseteq [-a, a]$.

The frontier of a closed box is the boundary of the box.

The frontier of a box is the boundary of the box.

If $c$ and $d$ are two points in $\mathbb{R}^n$ such that the box $[c,d]$ is nonempty, then the intersection of the box $[a,b]$ with the box $[c,d]$ is empty if and only if the intersection of the box $[a,b]$ with the box $[c,d]$ is empty.

If the range of a sequence of vectors is bounded, then for every subset $d$ of the basis of the vector space, there exists a vector $l$ and a strictly increasing sequence of natural numbers $r$ such that for every $\epsilon > 0$, there exists $n$ such that for all $i \in d$, we have $|f(r(n))_i - l_i| < \epsilon$.

The closed box $[a,b]$ is compact.

A set $S$ is an interval and compact if and only if there exist $a$ and $b$ such that $S = [a, b]$.

For any two points $x$ and $y$ in $\mathbb{R}^n$, the distance between the $i$th coordinates of $x$ and $y$ is less than or equal to the distance between $x$ and $y$.

If $S$ is an open set in $\mathbb{R}$, then the set $\{x \in \mathbb{R}^n \mid x_i \in S\}$ is open in $\mathbb{R}^n$.

A sequence of vectors $f_n$ converges to a vector $l$ if and only if each of its components converges to the corresponding component of $l$.

A function $f: \mathbb{R}^n \to \mathbb{R}^n$ is continuous if and only if each of its component functions is continuous.

A function $f$ is continuous on a set $S$ if and only if each of its components is continuous on $S$.

A function $f'$ is linear if and only if each of its components is linear.

A linear map $f'$ is bounded if and only if each of its components is bounded.

If $a$ and $b$ are vectors such that $a_i \leq b_i$ for all $i$, then the clamp function maps $x$ to a point in the box $[a, b]$.

If $x$ is in the closed box $[a,b]$, then clamping $x$ to $[a,b]$ does nothing.

If $a$ and $b$ are two points in $\mathbb{R}^n$ such that $a_i > b_i$ for some $i$, then the clamp function returns $a$ for all points.

The distance between two clamped points is less than or equal to the distance between the original points.

If $f$ is continuous on the closed box $[a, b]$, then the function $x \mapsto f(clamp(a, b, x))$ is continuous at $x$.

Suppose $f$ is a continuous function defined on the closed box $[a,b]$. Then the function $g$ defined by $g(x) = f(clamp(a,b,x))$ is continuous on any set $S$.

If $f$ is a function from a closed box to a metric space, then the function that maps each point in the box to the closest point in the image of $f$ is bounded.

If $f$ is a continuous function on a closed box $[a,b]$, then $f$ is equal to its extension by continuity.

If $f$ is continuous on the closed interval $[a,b]$, then the extension of $f$ to the whole real line is continuous on any set $S$.

There exists a sequence of open sets $B_n$ such that $B_n$ is injective and for any open set $S$, there exists a finite set $k$ such that $S = \bigcup_{n \in k} B_n$.

Every separable metric space is the closure of a countable subset.

The diameter of a closed ball is $0$ if the radius is negative, and $2r$ if the radius is nonnegative.