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

Statement: stringlengths 7 24.3k
The diameter of a ball is $2r$ if $r > 0$ and $0$ if $r \leq 0$.
The diameter of a closed interval is the difference between the endpoints, unless the interval is empty.
The diameter of an open interval is the length of the interval.
If $a$ and $b$ are two points in $\mathbb{R}^n$ such that $a_i \leq b_i$ for all $i$, then the diameter of the box with corners $a$ and $b$ is equal to the distance between $a$ and $b$.
If $f$ is a linear surjection from $\mathbb{R}^n$ to $\mathbb{R}^m$, then the image of an open set under $f$ is open.
If $f$ is a linear bijection, then $f$ maps open sets to open sets.
If $f$ is a linear bijection, then the interior of the image of a set $S$ is the image of the interior of $S$.
If $f$ is a linear injection, then the interior of the image of a set $S$ is the image of the interior of $S$.
If $f$ is a linear surjective map, then the interior of the image of a set $S$ is the image of the interior of $S$.
The interior of the set of negations of a set $S$ is the set of negations of the interior of $S$.
If $f$ is a linear map and $s$ is a connected set, then $f(s)$ is connected.
If $f$ is a linear map from a subspace $S$ of a Euclidean space to another Euclidean space, and $f$ is injective on $S$, then $f$ is bounded below on $S$.
If $f$ is a linear map from a closed subspace $s$ of $\mathbb{R}^n$ to $\mathbb{R}^m$ such that $f(x) = 0$ implies $x = 0$, then the image of $s$ under $f$ is closed.
If $f$ is a bounded linear map, then the image of the closure of a set $S$ is a subset of the closure of the image of $S$.
If $f$ is a linear map, then the image of the closure of a set $S$ is contained in the closure of the image of $S$.
If $f$ is a linear injection from a Euclidean space to another Euclidean space, then the image of a closed set under $f$ is closed.
If $f$ is a linear injective map, then $f$ maps closed sets to closed sets.
If $f$ is a linear injection, then $f$ maps the closure of a set $S$ to the closure of $f(S)$.
If $f$ is a linear map and $S$ is a bounded set, then the image of the closure of $S$ under $f$ is the closure of the image of $S$ under $f$.
The closure of the image of a set under a scalar multiplication is the image of the closure of the set under the scalar multiplication.
The set of points in $\mathbb{R}^n$ that are zero in all coordinates where $P$ is true is closed.
Any subspace of a Euclidean space is closed.
Any subspace of a complete space is complete.
The span of a set is closed.
The dimension of the closure of a set is equal to the dimension of the set.
If $A$ is compact and $B$ is closed, then there exist $x \in A$ and $y \in B$ such that $d(x,y) = d(A,B)$.
If $S$ is a closed set and $T$ is a compact set, then there exist $x \in S$ and $y \in T$ such that $d(x,y) = d(S,T)$.
If $S$ is compact and $T$ is closed, then $d(S,T) = 0$ if and only if $S$ and $T$ have a nonempty intersection.
If $S$ and $T$ are compact and closed, respectively, then $d(S,T) > 0$ if and only if $S \neq \emptyset$, $T \neq \emptyset$, and $S \cap T = \emptyset$.
If $S$ is a closed set and $T$ is a compact set, then $d(S,T) = 0$ if and only if $S$ and $T$ are nonempty and have a nonempty intersection.
If either $S$ or $T$ is bounded, then $d(S,T) = 0$ if and only if $S$ and $T$ are either both empty or have nonempty intersection.
The distance between a singleton set and a set $S$ is zero if and only if $S$ is empty or $x$ is in the closure of $S$.
The distance between a set $S$ and a point $x$ is zero if and only if $S$ is empty or $x$ is in the closure of $S$.
If the distance between a point $x$ and a set $S$ is nonzero, then $S$ is nonempty and $x$ is not in the closure of $S$.
If the distance between a set $S$ and a point $x$ is nonzero, then $S$ is nonempty and $x$ is not in the closure of $S$.
If $x$ is in $S$, then the distance between the singleton set $\{x\}$ and $S$ is $0$.
If $S$ is a closed set, then $d(x, S) = 0$ if and only if $S = \emptyset$ or $x \in S$.
If $S$ is closed in $U$ and $x \in U$, then $d(x,S) = 0$ if and only if $S = \emptyset$ or $x \in S$.
If $S$ is a closed subset of $U$ and $x \in U$ but $x \notin S$, then the distance between $x$ and $S$ is positive.
If $0 \leq x \leq 1$, then the Bernstein polynomial $B_{n,k}(x)$ is nonnegative.
If $0 < x < 1$ and $k \leq n$, then the Bernstein polynomial $B_{n,k}(x)$ is positive.
The sum of the Bernstein polynomials of degree $n$ is $1$.
The derivative of the binomial theorem is the binomial theorem.
The derivative of the binomial theorem is equal to the binomial theorem.
The sum of the $k$-th Bernstein polynomials for $k = 0, \ldots, n$ is $nx$.
The sum of the $k$th Bernstein polynomial of degree $n$ over $k$ from $0$ to $n$ is $n(n-1)x^2$.
Suppose $f$ is a continuous real-valued function defined on the interval $[0,1]$. For every $\epsilon > 0$, there exists a positive integer $N$ such that for all $n \geq N$ and all $x \in [0,1]$, we have $\|f(x) - \sum_{k=0}^n f(k/n) B_n^k(x)\| < \epsilon$.
If $f$ is a real-valued function, then $-f$ is also a real-valued function.
If $f$ and $g$ are Riemann integrable, then so is $f - g$.
If $f$ is a real-valued function, then $x \mapsto f(x)^n$ is also a real-valued function.
If $f$ is a function from a finite set $I$ to a ring $R$, then the function $g$ defined by $g(x) = \sum_{i \in I} f(i)(x)$ is also in $R$.
If $f$ is a function from a finite set $I$ to a ring $R$, then the function $x \mapsto \prod_{i \in I} f(i)$ is in $R$.
If $f$ is a continuous function defined on a set $S$, then for any $x \in S$, we have $\|f(x)\| \leq \\|f\\|$.
If $f$ is a function from a nonempty set $S$ to the real numbers, and if $f$ is bounded above by $M$, then the norm of $f$ is less than or equal to $M$.
Let $S$ be a topological space and $U$ be an open subset of $S$. If $t_0 \in U$ and $t_1 \in S - U$, then there exists an open set $V$ such that $t_0 \in V$, $S \cap V \subseteq U$, and for every $\epsilon > 0$, there exists a function $f \in R$ such that $f(S) \subseteq [0, 1]$, $f(t) < \epsilon$ for all $t \in S \cap V$, and $f(t) > 1 - \epsilon$ for all $t \in S - U$.
If $A$ and $B$ are disjoint closed subsets of a compact Hausdorff space $S$, and $a \in A$ and $b \in B$, then there exists a continuous function $f : S \to [0,1]$ such that $f(a) < \epsilon$ and $f(b) > 1 - \epsilon$ for some $\epsilon > 0$.
If $A$ and $B$ are disjoint closed subsets of $S$, then there exists a continuous function $f$ from $S$ to $[0,1]$ such that $f(A) \subseteq [0,\epsilon)$ and $f(B) \subseteq (1-\epsilon,1]$.
Suppose $f$ is a continuous real-valued function defined on a compact set $S$ such that $f(x) \geq 0$ for all $x \in S$. For every $\epsilon > 0$ such that $\epsilon < 1/3$, there exists a function $g$ in the ring $R$ such that for all $x \in S$, we have $\|f(x) - g(x)\| < 2\epsilon$.
Suppose $f$ is a continuous real-valued function defined on a compact set $S$. For every $\epsilon > 0$, there exists a polynomial p such that for all $x \in S$, we have $\|f(x) - p(x)\| < \epsilon$.
If $f$ is a continuous function on a compact set $S$, then there exists a sequence of polynomials $p_n$ such that $p_n$ converges uniformly to $f$ on $S$.
If $f$ is a continuous real-valued function defined on a compact set $S$, then for every $\epsilon > 0$, there exists a function $g$ in the set $R$ such that for all $x \in S$, we have $\|f(x) - g(x)\| < \epsilon$.
A real-valued polynomial function is the same as a polynomial function.
The constant function $f(x) = c$ is a polynomial function.
Any bounded linear function is a polynomial function.
The identity function is a polynomial function.
If $f$ and $g$ are polynomial functions, then so is $f + g$.
If $f$ and $g$ are polynomial functions, then so is $f \cdot g$.
If $f$ is a polynomial function, then so is $c \cdot f$.
If $f$ is a polynomial function, then so is $-f$.
If $f$ and $g$ are polynomial functions, then so is $f - g$.
If $f_i$ is a polynomial function for each $i \in I$, then $\sum_{i \in I} f_i$ is a polynomial function.
If $f$ is a real-valued polynomial function, then so is $-f$.
If $f$ and $g$ are real polynomial functions, then so is $f - g$.
If $f_i$ is a polynomial function for each $i \in I$, then the function $x \mapsto \sum_{i \in I} f_i(x)$ is a polynomial function.
If $f$ is a real polynomial function, then $f^n$ is a real polynomial function.
If $f$ is a polynomial function and $g$ is a real-valued polynomial function, then $g \circ f$ is a real-valued polynomial function.
If $f$ and $g$ are polynomial functions, then so is $g \circ f$.
For any real number $x$ and any two natural numbers $m$ and $n$, the sum of $x^i$ times $a_i$ for $i \leq m$ is equal to the sum of $x^i$ times $a_i$ for $i \leq \max(m,n)$ if $a_i = 0$ for $i > m$.
If $f$ is a real polynomial function, then there exist $a_0, a_1, \ldots, a_n \in \mathbb{R}$ such that $f(x) = a_0 + a_1 x + \cdots + a_n x^n$.
A real-valued function $f$ is a polynomial function if and only if there exist real numbers $a_0, a_1, \ldots, a_n$ such that $f(x) = a_0 + a_1 x + \cdots + a_n x^n$.
A function $f$ is a polynomial function if and only if the inner product of $f$ with every basis vector is a real-valued polynomial function.
If $f$ is a real polynomial function, then $f$ is continuous at $x$.
If $f$ is a polynomial function, then $f$ is continuous at $x$.
If $f$ is a polynomial function, then $f$ is continuous on $S$.
If $p$ is a real polynomial function, then there exists a real polynomial function $p'$ such that $p'$ is the derivative of $p$.
If $p$ is a polynomial function, then there exists a polynomial function $p'$ such that $p'$ is the derivative of $p$.
If $x$ and $y$ are distinct points in $\mathbb{R}^n$, then there exists a polynomial function $f$ such that $f(x) \neq f(y)$.
Suppose $f$ is a continuous real-valued function defined on a compact set $S$. For every $\epsilon > 0$, there exists a real polynomial $p$ such that for all $x \in S$, we have $\|f(x) - p(x)\| < \epsilon$.
Suppose $f$ is a continuous function defined on a compact set $S$. For every $\epsilon > 0$, there exists a polynomial p such that for all $x \<in> S$, we have $\|f(x) - p(x)\| < \epsilon$.
Suppose $f$ is a continuous function defined on a compact set $S$. Then there exists a sequence of polynomial functions $g_n$ such that $g_n$ converges uniformly to $f$ on $S$.
If $g$ is a polynomial function, then $g$ is a path.
Suppose $g$ is a path in $\mathbb{R}^n$. For every $\epsilon > 0$, there exists a polynomial $p$ such that $p(0) = g(0)$, $p(1) = g(1)$, and for all $t \in [0,1]$, we have $\|p(t) - g(t)\| < \epsilon$.
If $S$ is an open connected set in $\mathbb{R}^n$, then for any two points $x, y \in S$, there exists a polynomial function $g$ such that $g$ maps the interval $[0, 1]$ into $S$ and $g(0) = x$ and $g(1) = y$.
A function $f$ is differentiable at a point $a$ within a set $S$ if and only if each component function $f_i$ is differentiable at $a$ within $S$.
If $g$ is a polynomial function, then so is the inner product of $g$ with any fixed vector $i$.
If $f$ is a real polynomial function, then $f$ is differentiable at $a$ within $S$.
If $p$ is a real polynomial function, then $p$ is differentiable on $S$.
If $f$ is a polynomial function, then $f$ is differentiable at $a$ within $S$.
If $f$ is a polynomial function, then $f$ is differentiable on $S$.
If $x$ and $y$ are in the span of $B$ and $i \bullet x = i \bullet y$ for all $i \in B$, then $x = y$.

The diameter of a ball is $2r$ if $r > 0$ and $0$ if $r \leq 0$.

The diameter of a closed interval is the difference between the endpoints, unless the interval is empty.

The diameter of an open interval is the length of the interval.

If $a$ and $b$ are two points in $\mathbb{R}^n$ such that $a_i \leq b_i$ for all $i$, then the diameter of the box with corners $a$ and $b$ is equal to the distance between $a$ and $b$.

If $f$ is a linear surjection from $\mathbb{R}^n$ to $\mathbb{R}^m$, then the image of an open set under $f$ is open.

If $f$ is a linear bijection, then $f$ maps open sets to open sets.

If $f$ is a linear bijection, then the interior of the image of a set $S$ is the image of the interior of $S$.

If $f$ is a linear injection, then the interior of the image of a set $S$ is the image of the interior of $S$.

If $f$ is a linear surjective map, then the interior of the image of a set $S$ is the image of the interior of $S$.

The interior of the set of negations of a set $S$ is the set of negations of the interior of $S$.

If $f$ is a linear map and $s$ is a connected set, then $f(s)$ is connected.

If $f$ is a linear map from a subspace $S$ of a Euclidean space to another Euclidean space, and $f$ is injective on $S$, then $f$ is bounded below on $S$.

If $f$ is a linear map from a closed subspace $s$ of $\mathbb{R}^n$ to $\mathbb{R}^m$ such that $f(x) = 0$ implies $x = 0$, then the image of $s$ under $f$ is closed.

If $f$ is a bounded linear map, then the image of the closure of a set $S$ is a subset of the closure of the image of $S$.

If $f$ is a linear map, then the image of the closure of a set $S$ is contained in the closure of the image of $S$.

If $f$ is a linear injection from a Euclidean space to another Euclidean space, then the image of a closed set under $f$ is closed.

If $f$ is a linear injective map, then $f$ maps closed sets to closed sets.

If $f$ is a linear injection, then $f$ maps the closure of a set $S$ to the closure of $f(S)$.

If $f$ is a linear map and $S$ is a bounded set, then the image of the closure of $S$ under $f$ is the closure of the image of $S$ under $f$.

The closure of the image of a set under a scalar multiplication is the image of the closure of the set under the scalar multiplication.

The set of points in $\mathbb{R}^n$ that are zero in all coordinates where $P$ is true is closed.

Any subspace of a Euclidean space is closed.

Any subspace of a complete space is complete.

The span of a set is closed.

The dimension of the closure of a set is equal to the dimension of the set.

If $A$ is compact and $B$ is closed, then there exist $x \in A$ and $y \in B$ such that $d(x,y) = d(A,B)$.

If $S$ is a closed set and $T$ is a compact set, then there exist $x \in S$ and $y \in T$ such that $d(x,y) = d(S,T)$.

If $S$ is compact and $T$ is closed, then $d(S,T) = 0$ if and only if $S$ and $T$ have a nonempty intersection.

If $S$ and $T$ are compact and closed, respectively, then $d(S,T) > 0$ if and only if $S \neq \emptyset$, $T \neq \emptyset$, and $S \cap T = \emptyset$.

If $S$ is a closed set and $T$ is a compact set, then $d(S,T) = 0$ if and only if $S$ and $T$ are nonempty and have a nonempty intersection.

If either $S$ or $T$ is bounded, then $d(S,T) = 0$ if and only if $S$ and $T$ are either both empty or have nonempty intersection.

The distance between a singleton set and a set $S$ is zero if and only if $S$ is empty or $x$ is in the closure of $S$.

The distance between a set $S$ and a point $x$ is zero if and only if $S$ is empty or $x$ is in the closure of $S$.

If the distance between a point $x$ and a set $S$ is nonzero, then $S$ is nonempty and $x$ is not in the closure of $S$.

If the distance between a set $S$ and a point $x$ is nonzero, then $S$ is nonempty and $x$ is not in the closure of $S$.

If $x$ is in $S$, then the distance between the singleton set $\{x\}$ and $S$ is $0$.

If $S$ is a closed set, then $d(x, S) = 0$ if and only if $S = \emptyset$ or $x \in S$.

If $S$ is closed in $U$ and $x \in U$, then $d(x,S) = 0$ if and only if $S = \emptyset$ or $x \in S$.

If $S$ is a closed subset of $U$ and $x \in U$ but $x \notin S$, then the distance between $x$ and $S$ is positive.

If $0 \leq x \leq 1$, then the Bernstein polynomial $B_{n,k}(x)$ is nonnegative.

If $0 < x < 1$ and $k \leq n$, then the Bernstein polynomial $B_{n,k}(x)$ is positive.

The sum of the Bernstein polynomials of degree $n$ is $1$.

The derivative of the binomial theorem is the binomial theorem.

The derivative of the binomial theorem is equal to the binomial theorem.

The sum of the $k$-th Bernstein polynomials for $k = 0, \ldots, n$ is $nx$.

The sum of the $k$th Bernstein polynomial of degree $n$ over $k$ from $0$ to $n$ is $n(n-1)x^2$.

Suppose $f$ is a continuous real-valued function defined on the interval $[0,1]$. For every $\epsilon > 0$, there exists a positive integer $N$ such that for all $n \geq N$ and all $x \in [0,1]$, we have $|f(x) - \sum_{k=0}^n f(k/n) B_n^k(x)| < \epsilon$.

If $f$ is a real-valued function, then $-f$ is also a real-valued function.

If $f$ and $g$ are Riemann integrable, then so is $f - g$.

If $f$ is a real-valued function, then $x \mapsto f(x)^n$ is also a real-valued function.

If $f$ is a function from a finite set $I$ to a ring $R$, then the function $g$ defined by $g(x) = \sum_{i \in I} f(i)(x)$ is also in $R$.

If $f$ is a function from a finite set $I$ to a ring $R$, then the function $x \mapsto \prod_{i \in I} f(i)$ is in $R$.

If $f$ is a continuous function defined on a set $S$, then for any $x \in S$, we have $|f(x)| \leq \|f\|$.

If $f$ is a function from a nonempty set $S$ to the real numbers, and if $f$ is bounded above by $M$, then the norm of $f$ is less than or equal to $M$.

Let $S$ be a topological space and $U$ be an open subset of $S$. If $t_0 \in U$ and $t_1 \in S - U$, then there exists an open set $V$ such that $t_0 \in V$, $S \cap V \subseteq U$, and for every $\epsilon > 0$, there exists a function $f \in R$ such that $f(S) \subseteq [0, 1]$, $f(t) < \epsilon$ for all $t \in S \cap V$, and $f(t) > 1 - \epsilon$ for all $t \in S - U$.

If $A$ and $B$ are disjoint closed subsets of a compact Hausdorff space $S$, and $a \in A$ and $b \in B$, then there exists a continuous function $f : S \to [0,1]$ such that $f(a) < \epsilon$ and $f(b) > 1 - \epsilon$ for some $\epsilon > 0$.

If $A$ and $B$ are disjoint closed subsets of $S$, then there exists a continuous function $f$ from $S$ to $[0,1]$ such that $f(A) \subseteq [0,\epsilon)$ and $f(B) \subseteq (1-\epsilon,1]$.

Suppose $f$ is a continuous real-valued function defined on a compact set $S$ such that $f(x) \geq 0$ for all $x \in S$. For every $\epsilon > 0$ such that $\epsilon < 1/3$, there exists a function $g$ in the ring $R$ such that for all $x \in S$, we have $|f(x) - g(x)| < 2\epsilon$.

Suppose $f$ is a continuous real-valued function defined on a compact set $S$. For every $\epsilon > 0$, there exists a polynomial p such that for all $x \in S$, we have $|f(x) - p(x)| < \epsilon$.

If $f$ is a continuous function on a compact set $S$, then there exists a sequence of polynomials $p_n$ such that $p_n$ converges uniformly to $f$ on $S$.

If $f$ is a continuous real-valued function defined on a compact set $S$, then for every $\epsilon > 0$, there exists a function $g$ in the set $R$ such that for all $x \in S$, we have $|f(x) - g(x)| < \epsilon$.

A real-valued polynomial function is the same as a polynomial function.

The constant function $f(x) = c$ is a polynomial function.

Any bounded linear function is a polynomial function.

The identity function is a polynomial function.

If $f$ and $g$ are polynomial functions, then so is $f + g$.

If $f$ and $g$ are polynomial functions, then so is $f \cdot g$.

If $f$ is a polynomial function, then so is $c \cdot f$.

If $f$ is a polynomial function, then so is $-f$.

If $f$ and $g$ are polynomial functions, then so is $f - g$.

If $f_i$ is a polynomial function for each $i \in I$, then $\sum_{i \in I} f_i$ is a polynomial function.

If $f$ is a real-valued polynomial function, then so is $-f$.

If $f$ and $g$ are real polynomial functions, then so is $f - g$.

If $f_i$ is a polynomial function for each $i \in I$, then the function $x \mapsto \sum_{i \in I} f_i(x)$ is a polynomial function.

If $f$ is a real polynomial function, then $f^n$ is a real polynomial function.

If $f$ is a polynomial function and $g$ is a real-valued polynomial function, then $g \circ f$ is a real-valued polynomial function.

If $f$ and $g$ are polynomial functions, then so is $g \circ f$.

For any real number $x$ and any two natural numbers $m$ and $n$, the sum of $x^i$ times $a_i$ for $i \leq m$ is equal to the sum of $x^i$ times $a_i$ for $i \leq \max(m,n)$ if $a_i = 0$ for $i > m$.

If $f$ is a real polynomial function, then there exist $a_0, a_1, \ldots, a_n \in \mathbb{R}$ such that $f(x) = a_0 + a_1 x + \cdots + a_n x^n$.

A real-valued function $f$ is a polynomial function if and only if there exist real numbers $a_0, a_1, \ldots, a_n$ such that $f(x) = a_0 + a_1 x + \cdots + a_n x^n$.

A function $f$ is a polynomial function if and only if the inner product of $f$ with every basis vector is a real-valued polynomial function.

If $f$ is a real polynomial function, then $f$ is continuous at $x$.

If $f$ is a polynomial function, then $f$ is continuous at $x$.

If $f$ is a polynomial function, then $f$ is continuous on $S$.

If $p$ is a real polynomial function, then there exists a real polynomial function $p'$ such that $p'$ is the derivative of $p$.

If $p$ is a polynomial function, then there exists a polynomial function $p'$ such that $p'$ is the derivative of $p$.

If $x$ and $y$ are distinct points in $\mathbb{R}^n$, then there exists a polynomial function $f$ such that $f(x) \neq f(y)$.

Suppose $f$ is a continuous real-valued function defined on a compact set $S$. For every $\epsilon > 0$, there exists a real polynomial $p$ such that for all $x \in S$, we have $|f(x) - p(x)| < \epsilon$.

Suppose $f$ is a continuous function defined on a compact set $S$. For every $\epsilon > 0$, there exists a polynomial p such that for all $x \<in> S$, we have $|f(x) - p(x)| < \epsilon$.

Suppose $f$ is a continuous function defined on a compact set $S$. Then there exists a sequence of polynomial functions $g_n$ such that $g_n$ converges uniformly to $f$ on $S$.

If $g$ is a polynomial function, then $g$ is a path.

Suppose $g$ is a path in $\mathbb{R}^n$. For every $\epsilon > 0$, there exists a polynomial $p$ such that $p(0) = g(0)$, $p(1) = g(1)$, and for all $t \in [0,1]$, we have $|p(t) - g(t)| < \epsilon$.

If $S$ is an open connected set in $\mathbb{R}^n$, then for any two points $x, y \in S$, there exists a polynomial function $g$ such that $g$ maps the interval $[0, 1]$ into $S$ and $g(0) = x$ and $g(1) = y$.

A function $f$ is differentiable at a point $a$ within a set $S$ if and only if each component function $f_i$ is differentiable at $a$ within $S$.

If $g$ is a polynomial function, then so is the inner product of $g$ with any fixed vector $i$.

If $f$ is a real polynomial function, then $f$ is differentiable at $a$ within $S$.

If $p$ is a real polynomial function, then $p$ is differentiable on $S$.

If $f$ is a polynomial function, then $f$ is differentiable at $a$ within $S$.

If $f$ is a polynomial function, then $f$ is differentiable on $S$.

If $x$ and $y$ are in the span of $B$ and $i \bullet x = i \bullet y$ for all $i \in B$, then $x = y$.