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If $a$ and $b$ are real numbers, then $a + b$ is a real number. |
If $a$ is a real number, then $-a$ is a real number. |
$-a \in \mathbb{R}$ if and only if $a \in \mathbb{R}$. |
If $a$ and $b$ are real numbers, then $a - b$ is a real number. |
If $a$ and $b$ are real numbers, then $a \cdot b$ is a real number. |
If $a$ is a nonzero real number, then $1/a$ is a real number. |
If $a$ is a real number, then $1/a$ is a real number. |
The inverse of a real number is real if and only if the number is real. |
If $a$ and $b$ are real numbers and $b \neq 0$, then $a/b$ is a real number. |
If $a$ and $b$ are real numbers, then $a/b$ is a real number. |
If $a$ is a real number, then $a^n$ is a real number. |
If $q$ is a real number, then there exists a real number $r$ such that $q = r$. |
If $f$ is a function from a set $s$ to the set of real numbers, then the sum of $f$ over $s$ is a real number. |
If $f(i) \in \mathbb{R}$ for all $i \in s$, then $\prod_{i \in s} f(i) \in \mathbb{R}$. |
If $q$ is a real number and $P$ holds for all real numbers, then $P$ holds for $q$. |
If $a \leq b$ and $x \leq y$, and $a, b, x, y$ are all nonnegative, then $a x \leq b y$. |
If $a \leq b$ and $c \leq d$, and $a, c \geq 0$, then $ac \leq bd$. |
If $c > 0$, then $a \leq b/c$ if and only if $ca \leq b$. |
If $c > 0$, then $a < b/c$ if and only if $ca < b$. |
If $c > 0$, then $b/c \leq a$ if and only if $b \leq ca$. |
If $c > 0$, then $b/c < a$ if and only if $b < ca$. |
If $c > 0$, then $a \leq -\frac{b}{c}$ if and only if $c \cdot a \leq -b$. |
If $c > 0$, then $a < - \frac{b}{c}$ if and only if $c \cdot a < - b$. |
If $c > 0$, then $-\frac{b}{c} \leq a$ if and only if $-b \leq ca$. |
If $c > 0$, then $-\frac{b}{c} < a$ if and only if $-b < ca$. |
If $c > 0$, then the image of the interval $[x, y]$ under the scaling map $c \cdot$ is the interval $[cx, cy]$. |
If $c < 0$, then $a \leq b/c$ if and only if $b \leq ca$. |
If $c < 0$, then $a < b/c$ if and only if $b < ca$. |
If $c < 0$, then $b/c \leq a$ if and only if $c \cdot a \leq b$. |
If $c < 0$, then $b/c < a$ if and only if $ca < b$. |
If $c < 0$, then $a \leq - \frac{b}{c}$ if and only if $-b \leq ca$. |
If $c < 0$, then $a < - \frac{b}{c}$ if and only if $-b < ca$. |
If $c < 0$, then $-\frac{b}{c} \leq a$ if and only if $c \cdot a \leq -b$. |
If $c < 0$, then $-\frac{b}{c} < a$ if and only if $c \cdot a < -b$. |
The following equations hold for all real vectors $a$, $b$, and $c$: $a = b / c$ if and only if $c = 0$ or $c a = b$ $b / c = a$ if and only if $c = 0$ or $b = c a$ $a + b / c = (c a + b) / c$ if and only if $c \neq 0$ $a / c + b = (a + c b) / c$ if and only if $c \neq 0$ $a - b / c = (c a - b) / c$ if and only if $c \neq 0$ $a / c - b = (a - c b) / c$ if and only if $c \neq 0$ $- (a / c) + b = (- a + c b) / c$ if and only if $c \neq 0$ $- (a / c) - b = (- a - c b) / c$ if and only if $c \neq 0$ |
If $c > 0$, then $a \leq b/c$ if and only if $ca \leq b$. If $c < 0$, then $a \leq b/c$ if and only if $b \leq ca$. If $c = 0$, then $a \leq b/c$ if and only if $a \leq 0$. |
If $a$ and $x$ are nonnegative real numbers, then $a \cdot x$ is nonnegative. |
If $a \geq 0$ and $x \leq 0$, then $a \cdot x \leq 0$. |
If $a \leq 0$ and $x \geq 0$, then $a \cdot x \leq 0$. |
If $a$ and $x$ are real numbers such that either $a \geq 0$ and $x \leq 0$, or $a \leq 0$ and $x \geq 0$, then $a \cdot x \leq 0$. |
For any real vector $e$, we have $a e + c \leq b e + d$ if and only if $(a - b) e + c \leq d$. |
For any real vector $e$, we have $a e + c \leq b e + d$ if and only if $c \leq (b - a) e + d$. |
If $b \leq a$ and $c \leq 0$, then $c \cdot a \leq c \cdot b$. |
If $a \leq b$ and $c \leq 0$, then $ac \leq bc$. |
If $a$ and $b$ are non-positive real numbers, then $ab$ is non-negative. |
If $a$ and $b$ are both non-negative or both non-positive, then $a \cdot b$ is non-negative. |
For any real vector $b$, the inequality $0 \leq a b$ holds if and only if $0 < a$ and $0 \leq b$ or $a < 0$ and $b \leq 0$ or $a = 0$. |
For any real vector $b$, the product $ab$ is non-positive if and only if $a$ is positive and $b$ is non-positive, or $a$ is negative and $b$ is non-negative, or $a$ is zero. |
For any real vector $a$ and $b$, $c a \leq c b$ if and only if $a \leq b$ if $c > 0$ and $b \leq a$ if $c < 0$. |
If $c > 0$, then $c \cdot a \leq c \cdot b$ if and only if $a \leq b$. |
If $c < 0$, then $c \cdot a \leq c \cdot b$ if and only if $b \leq a$. |
If $x$ is a non-negative real number and $a$ is a real number between $0$ and $1$, then $a \cdot x \leq x$. |
A predicate $P$ holds eventually in the uniformity of a metric space if and only if there exists $\epsilon > 0$ such that $P$ holds for all pairs of points whose distance is less than $\epsilon$. |
The norm of a complex number is nonnegative. |
For any real number $x$ and any natural number $n$, $(x y)^n = x^n y^n$. |
The norm of the zero vector is zero. |
The norm of a complex number is positive if and only if the complex number is nonzero. |
The norm of a complex number is never less than zero. |
The norm of a complex number is less than or equal to zero if and only if the complex number is zero. |
The norm of a number is the same as the norm of its negative. |
The norm of a complex number is invariant under complex conjugation. |
The distance between $a + b$ and $a + c$ is the same as the distance between $b$ and $c$. |
The distance between $b + a$ and $c + a$ is the same as the distance between $b$ and $c$. |
The norm of $-x - y$ is equal to the norm of $x + y$. |
The norm of a complex number is always greater than or equal to the difference of the norms of its real and imaginary parts. |
The absolute value of the difference between the norms of two vectors is less than or equal to the norm of the difference between the vectors. |
The norm of the difference of two vectors is less than or equal to the sum of their norms. |
If the sum of the norms of two vectors is less than or equal to some number, then the norm of their difference is less than or equal to that number. |
$\|a\| - \|b\| \leq \|a + b\|$. |
The norm of $x$ is less than or equal to the norm of $y$ plus the norm of $x - y$. |
If $x$ and $y$ are vectors such that $||x|| + ||y|| \leq e$, then $||x + y|| \leq e$. |
If $|x| + |y| < \epsilon$, then $|x + y| < \epsilon$. |
If $\|a + b\| \leq c$, then $\|b\| \leq \|a\| + c$. |
The norm of the difference of two sums is less than or equal to the sum of the norms of the differences. |
If $x$ and $y$ are two real numbers such that $|x - y| \leq e_1$ and $|y - z| \leq e_2$, then $|x - z| \leq e_1 + e_2$. |
If $|x - y| < \epsilon_1$ and $|y - z| < \epsilon_2$, then $|x - z| < \epsilon_1 + \epsilon_2$. |
If $|a| \leq r$ and $|b| \leq s$, then $|a + b| \leq r + s$. |
The norm of a sum of vectors is less than or equal to the sum of the norms of the vectors. |
If $f$ is a function from a set $S$ to a normed vector space $V$, and $g$ is a function from $S$ to the nonnegative reals, then $\|\sum_{x \in S} f(x)\| \leq \sum_{x \in S} g(x)$ if $\|f(x)\| \leq g(x)$ for all $x \in S$. |
The absolute value of the norm of a complex number is equal to the norm of the complex number. |
If $f$ is a function from a finite set $S$ to a normed vector space $V$, and if $f(x)$ is bounded by $K$ for all $x \<in> S$, then $\sum_{x \in S} f(x)$ is bounded by $|S|K$. |
If $x$ and $y$ are complex numbers such that $|x| < r$ and $|y| < s$, then $|x + y| < r + s$. |
The distance between two scalar multiples of a vector is equal to the absolute value of the difference of the scalars times the norm of the vector. |
If $x$ and $y$ are complex numbers such that $|x| < r$ and $|y| < s$, then $|xy| < rs$. |
The norm of a real number is its absolute value. |
The norm of a numeral is the numeral itself. |
The norm of a negative numeral is the numeral itself. |
The norm of a real number plus 1 is the absolute value of the real number plus 1. |
The norm of a real number plus a numeral is the absolute value of the real number plus the numeral. |
The norm of an integer is the absolute value of the integer. |
The norm of a natural number is the natural number itself. |
If $a \neq 0$, then $\|a^{-1}\| = \|a\|^{-1}$. |
The norm of the inverse of a number is the inverse of the norm of the number. |
If $b \neq 0$, then $\|a/b\| = \|a\|/\|b\|$. |
The norm of a quotient is the quotient of the norms. |
If $x$ is a nonzero real number and $r \leq |x|$, then $|1/x| \leq 1/r$. |
For any real normed algebra $A$, and any $x \in A$, we have $||x^n|| \leq ||x||^n$. |
The norm of a power of a number is the power of the norm of the number. |
For any complex number $x$ and any integer $n$, we have $|x^n| = |x|^n$. |
If $w^n = z^n$ for some $n > 0$, then $|w| = |z|$. |