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

Statement: stringlengths 7 24.3k
If $w^n = 1$, then either $w$ has norm 1 or $n = 0$.
For any real-valued normed field $a$ and any natural number $n$, we have $\|n \cdot a\| = n \cdot \|a\|$.
For any real-valued field $a$ and any natural number $n$, $\|a \cdot n\| = \|a\| \cdot n$.
The norm of a quotient of two numerals is the quotient of their norms.
The norm of the difference of two real numbers is less than or equal to the absolute value of the difference of the two real numbers.
If $x^2 = 1$, then $\|x\| = 1$.
For any real normed algebra, the norm of any element is less than the norm of the element plus 1.
The norm of a product of numbers is equal to the product of the norms of the numbers.
The norm of the product of a family of elements is less than or equal to the product of the norms of the elements.
If $z_i$ and $w_i$ are vectors with norm at most 1, then the norm of the difference of their products is at most the sum of the norms of their differences.
If $z$ and $w$ are complex numbers with $\|z\| \leq 1$ and $\|w\| \leq 1$, then $\|z^m - w^m\| \leq m \|z - w\|$.
The distance between a point and itself is zero.
The distance between two points is always non-negative.
$0 < \text{dist}(x, y)$ if and only if $x \neq y$.
The distance between two points is never less than zero.
The distance between two points is non-negative and is zero if and only if the points are equal.
The distance between two points is the same as the distance between those points in the opposite order.
The distance between two points is the same in either direction.
The distance between $x$ and $z$ is less than or equal to the distance between $x$ and $y$ plus the distance between $y$ and $z$.
The distance between two points is less than or equal to the sum of the distances from each point to a third point.
The absolute value of the difference between the distance from $a$ to $b$ and the distance from $b$ to $c$ is less than or equal to the distance from $a$ to $c$.
If $x \neq y$, then $0 < \\|x - y\\|$.
$x \neq y$ if and only if $0 < \text{dist}(x, y)$.
The distance between two points is less than or equal to the sum of the distances from each point to a third point.
If the distance between $x$ and $z$ plus the distance between $y$ and $z$ is less than $e$, then the distance between $x$ and $y$ is less than $e$.
If $x_1$ is close to $y$ and $x_2$ is close to $y$, then $x_1$ is close to $x_2$.
If $x_1$ and $x_2$ are within $\frac{e}{2}$ of $y$, then $x_1$ and $x_2$ are within $e$ of each other.
If $y$ is within distance $e/2$ of both $x_1$ and $x_2$, then $x_1$ and $x_2$ are within distance $e$ of each other.
If the distance between $x_1$ and $x_2$ is less than $\frac{e}{3}$, the distance between $x_2$ and $x_3$ is less than $\frac{e}{3}$, and the distance between $x_3$ and $x_4$ is less than $\frac{e}{3}$, then the distance between $x_1$ and $x_4$ is less than $e$.
A set $S$ is open if and only if for every $x \in S$, there exists an $\epsilon > 0$ such that for all $y$, if $\|y - x\| < \epsilon$, then $y \in S$.
The open ball of radius $d$ around $x$ is open.
The distance between two real numbers is the same as the distance between their real-algebraic counterparts.
The set of real numbers greater than $a$ is open.
The set of real numbers less than a given real number is open.
The set of real numbers between two real numbers is open.
The set of real numbers less than or equal to a given real number is closed.
The set of real numbers greater than or equal to a given real number is closed.
The set of real numbers between $a$ and $b$ is closed.
The norm of the sign function is $0$ if $x = 0$ and $1$ otherwise.
The sign of zero is zero.
The sign function is zero if and only if the argument is zero.
The sign of a negative number is the negative of the sign of the number.
The sign of a scalar multiple of a vector is the scalar multiple of the signs of the scalar and the vector.
The sign of 1 is 1.
The sign of a real number is the same as the sign of its image under the embedding of the reals into any real normed algebra.
The sign of a product is the product of the signs.
The sign function is the quotient of $x$ and its absolute value.
For any real number $x$, $0 \leq \text{sgn}(x)$ if and only if $0 \leq x$.
For any real number $x$, $sgn(x) \leq 0$ if and only if $x \leq 0$.
The norm of a vector is equal to the distance from the origin.
The distance from $0$ to $x$ is equal to the norm of $x$.
The distance between $a$ and $a - b$ is equal to the norm of $b$.
The distance between two integers is the absolute value of their difference.
The distance between two natural numbers is the absolute value of their difference.
If $f$ is a function from a vector space to itself such that $f(b_1 + b_2) = f(b_1) + f(b_2)$ and $f(r b) = r f(b)$ for all $b_1, b_2 \in V$ and $r \in \mathbb{R}$, then $f$ is linear.
A function $f$ is linear if and only if it satisfies the two conditions: $f(x + y) = f(x) + f(y)$ $f(cx) = cf(x)$
The function $x \mapsto a x$ is linear.
If $f$ is a linear function, then $f$ is a scalar multiplication.
If $f$ is a linear function from $\mathbb{R}$ to $\mathbb{R}$, then there exists a real number $c$ such that $f(x) = cx$ for all $x \<in> \mathbb{R}$.
The function $x \mapsto a \cdot \mathrm{of\_real}(x)$ is linear.
There exists a constant $K$ such that for all $x$, $\|f(x)\| \leq \|x\|K$.
There exists a nonnegative real number $K$ such that for all $x$, we have $\|f(x)\| \leq \|x\|K$.
$f$ is a linear function.
If $f$ is a linear map such that $\\|f(x)\\| \leq K \\|x\\|$ for all $x$, then $f$ is bounded.
There exists a constant $K > 0$ such that for all matrices $a$ and $b$, we have $\\|ab\\| \leq \\|a\\| \\|b\\| K$.
There exists a constant $K \geq 0$ such that for all matrices $a$ and $b$, we have $\\|ab\\| \leq \\|a\\| \\|b\\| K$.
The function $b \mapsto a \cdot b$ is additive.
The function $a \mapsto a \cdot b$ is additive.
The product of $0$ and $b$ is $0$.
The product of any number and zero is zero.
The product of a negative number and a positive number is negative.
The product of a number and the negative of another number is the negative of the product of the two numbers.
$(a - a')b = ab - a'b$.
$a(b - b') = ab - ab'$.
The product of the sum of a family of functions is equal to the sum of the products of the functions.
$\prod_{x \in X} \sum_{y \in Y} f(x, y) = \sum_{y \in Y} \prod_{x \in X} f(x, y)$
The map $a \mapsto a \cdot b$ is a bounded linear operator.
The map $b \mapsto a \cdot b$ is a bounded linear operator.
$(x \cdot y - a \cdot b) = (x - a) \cdot (y - b) + (x - a) \cdot b + a \cdot (y - b)$
The function $(x, y) \mapsto y \cdot x$ is a bounded bilinear function.
If $g$ is a bounded linear operator, then the function $x \mapsto g(x)$ is a bounded bilinear operator.
If $f$ and $g$ are bounded linear maps, then the map $(x, y) \mapsto f(x) \cdot g(y)$ is a bounded bilinear map.
The identity function is a bounded linear operator.
The function $f(x) = 0$ is bounded linear.
If $f$ and $g$ are bounded linear maps, then so is $f + g$.
If $f$ is a bounded linear operator, then $-f$ is a bounded linear operator.
If $f$ and $g$ are bounded linear maps, then so is $f - g$.
If $f_i$ is a bounded linear map for each $i \in I$, then $\sum_{i \in I} f_i$ is a bounded linear map.
If $f$ and $g$ are bounded linear maps, then so is $f \circ g$.
The multiplication operation is bounded bilinear.
The function $x \mapsto xy$ is bounded linear.
The multiplication by a fixed element $x$ is a bounded linear operator.
The multiplication by a constant is a bounded linear operator.
The function $f(x) = cx$ is a bounded linear operator.
The function $x \mapsto x/y$ is a bounded linear operator.
The function $f(x, y) = xy$ is a bounded bilinear function.
The function $r \mapsto rx$ is a bounded linear operator.
The function $x \mapsto rx$ is a bounded linear operator.
The function $x \mapsto c x$ is a bounded linear operator on $\mathbb{R}$.
The function $x \mapsto c x$ is a bounded linear operator on $\mathbb{R}$.

If $w^n = 1$, then either $w$ has norm 1 or $n = 0$.

For any real-valued normed field $a$ and any natural number $n$, we have $|n \cdot a| = n \cdot |a|$.

For any real-valued field $a$ and any natural number $n$, $|a \cdot n| = |a| \cdot n$.

The norm of a quotient of two numerals is the quotient of their norms.

The norm of the difference of two real numbers is less than or equal to the absolute value of the difference of the two real numbers.

If $x^2 = 1$, then $|x| = 1$.

For any real normed algebra, the norm of any element is less than the norm of the element plus 1.

The norm of a product of numbers is equal to the product of the norms of the numbers.

The norm of the product of a family of elements is less than or equal to the product of the norms of the elements.

If $z_i$ and $w_i$ are vectors with norm at most 1, then the norm of the difference of their products is at most the sum of the norms of their differences.

If $z$ and $w$ are complex numbers with $|z| \leq 1$ and $|w| \leq 1$, then $|z^m - w^m| \leq m |z - w|$.

The distance between a point and itself is zero.

The distance between two points is always non-negative.

$0 < \text{dist}(x, y)$ if and only if $x \neq y$.

The distance between two points is never less than zero.

The distance between two points is non-negative and is zero if and only if the points are equal.

The distance between two points is the same as the distance between those points in the opposite order.

The distance between two points is the same in either direction.

The distance between $x$ and $z$ is less than or equal to the distance between $x$ and $y$ plus the distance between $y$ and $z$.

The distance between two points is less than or equal to the sum of the distances from each point to a third point.

The absolute value of the difference between the distance from $a$ to $b$ and the distance from $b$ to $c$ is less than or equal to the distance from $a$ to $c$.

If $x \neq y$, then $0 < \|x - y\|$.

$x \neq y$ if and only if $0 < \text{dist}(x, y)$.

The distance between two points is less than or equal to the sum of the distances from each point to a third point.

If the distance between $x$ and $z$ plus the distance between $y$ and $z$ is less than $e$, then the distance between $x$ and $y$ is less than $e$.

If $x_1$ is close to $y$ and $x_2$ is close to $y$, then $x_1$ is close to $x_2$.

If $x_1$ and $x_2$ are within $\frac{e}{2}$ of $y$, then $x_1$ and $x_2$ are within $e$ of each other.

If $y$ is within distance $e/2$ of both $x_1$ and $x_2$, then $x_1$ and $x_2$ are within distance $e$ of each other.

If the distance between $x_1$ and $x_2$ is less than $\frac{e}{3}$, the distance between $x_2$ and $x_3$ is less than $\frac{e}{3}$, and the distance between $x_3$ and $x_4$ is less than $\frac{e}{3}$, then the distance between $x_1$ and $x_4$ is less than $e$.

A set $S$ is open if and only if for every $x \in S$, there exists an $\epsilon > 0$ such that for all $y$, if $|y - x| < \epsilon$, then $y \in S$.

The open ball of radius $d$ around $x$ is open.

The distance between two real numbers is the same as the distance between their real-algebraic counterparts.

The set of real numbers greater than $a$ is open.

The set of real numbers less than a given real number is open.

The set of real numbers between two real numbers is open.

The set of real numbers less than or equal to a given real number is closed.

The set of real numbers greater than or equal to a given real number is closed.

The set of real numbers between $a$ and $b$ is closed.

The norm of the sign function is $0$ if $x = 0$ and $1$ otherwise.

The sign of zero is zero.

The sign function is zero if and only if the argument is zero.

The sign of a negative number is the negative of the sign of the number.

The sign of a scalar multiple of a vector is the scalar multiple of the signs of the scalar and the vector.

The sign of 1 is 1.

The sign of a real number is the same as the sign of its image under the embedding of the reals into any real normed algebra.

The sign of a product is the product of the signs.

The sign function is the quotient of $x$ and its absolute value.

For any real number $x$, $0 \leq \text{sgn}(x)$ if and only if $0 \leq x$.

For any real number $x$, $sgn(x) \leq 0$ if and only if $x \leq 0$.

The norm of a vector is equal to the distance from the origin.

The distance from $0$ to $x$ is equal to the norm of $x$.

The distance between $a$ and $a - b$ is equal to the norm of $b$.

The distance between two integers is the absolute value of their difference.

The distance between two natural numbers is the absolute value of their difference.

If $f$ is a function from a vector space to itself such that $f(b_1 + b_2) = f(b_1) + f(b_2)$ and $f(r b) = r f(b)$ for all $b_1, b_2 \in V$ and $r \in \mathbb{R}$, then $f$ is linear.

A function $f$ is linear if and only if it satisfies the two conditions: $f(x + y) = f(x) + f(y)$ $f(cx) = cf(x)$

The function $x \mapsto a x$ is linear.

If $f$ is a linear function, then $f$ is a scalar multiplication.

If $f$ is a linear function from $\mathbb{R}$ to $\mathbb{R}$, then there exists a real number $c$ such that $f(x) = cx$ for all $x \<in> \mathbb{R}$.

The function $x \mapsto a \cdot \mathrm{of\_real}(x)$ is linear.

There exists a constant $K$ such that for all $x$, $|f(x)| \leq |x|K$.

There exists a nonnegative real number $K$ such that for all $x$, we have $|f(x)| \leq |x|K$.

$f$ is a linear function.

If $f$ is a linear map such that $\|f(x)\| \leq K \|x\|$ for all $x$, then $f$ is bounded.

There exists a constant $K > 0$ such that for all matrices $a$ and $b$, we have $\|ab\| \leq \|a\| \|b\| K$.

There exists a constant $K \geq 0$ such that for all matrices $a$ and $b$, we have $\|ab\| \leq \|a\| \|b\| K$.

The function $b \mapsto a \cdot b$ is additive.

The function $a \mapsto a \cdot b$ is additive.

The product of $0$ and $b$ is $0$.

The product of any number and zero is zero.

The product of a negative number and a positive number is negative.

The product of a number and the negative of another number is the negative of the product of the two numbers.

$(a - a')b = ab - a'b$.

$a(b - b') = ab - ab'$.

The product of the sum of a family of functions is equal to the sum of the products of the functions.

$\prod_{x \in X} \sum_{y \in Y} f(x, y) = \sum_{y \in Y} \prod_{x \in X} f(x, y)$

The map $a \mapsto a \cdot b$ is a bounded linear operator.

The map $b \mapsto a \cdot b$ is a bounded linear operator.

$(x \cdot y - a \cdot b) = (x - a) \cdot (y - b) + (x - a) \cdot b + a \cdot (y - b)$

The function $(x, y) \mapsto y \cdot x$ is a bounded bilinear function.

If $g$ is a bounded linear operator, then the function $x \mapsto g(x)$ is a bounded bilinear operator.

If $f$ and $g$ are bounded linear maps, then the map $(x, y) \mapsto f(x) \cdot g(y)$ is a bounded bilinear map.

The identity function is a bounded linear operator.

The function $f(x) = 0$ is bounded linear.

If $f$ and $g$ are bounded linear maps, then so is $f + g$.

If $f$ is a bounded linear operator, then $-f$ is a bounded linear operator.

If $f$ and $g$ are bounded linear maps, then so is $f - g$.

If $f_i$ is a bounded linear map for each $i \in I$, then $\sum_{i \in I} f_i$ is a bounded linear map.

If $f$ and $g$ are bounded linear maps, then so is $f \circ g$.

The multiplication operation is bounded bilinear.

The function $x \mapsto xy$ is bounded linear.

The multiplication by a fixed element $x$ is a bounded linear operator.

The multiplication by a constant is a bounded linear operator.

The function $f(x) = cx$ is a bounded linear operator.

The function $x \mapsto x/y$ is a bounded linear operator.

The function $f(x, y) = xy$ is a bounded bilinear function.

The function $r \mapsto rx$ is a bounded linear operator.

The function $x \mapsto rx$ is a bounded linear operator.

The function $x \mapsto c x$ is a bounded linear operator on $\mathbb{R}$.