명제와 조건

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$, $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$. Complete each of the following sentences using the appropriate symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is not strictly increasing provided that $\ldots$.

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$, $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$. Complete each of the following sentences using the appropriate symbols for quantifiers:

Complete the following sentence in English without using symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is not strictly increasing provided that $\ldots$.

The following exercises contain definitions or results from more advanced mathematics courses. Even though we may not understand all of the terms involved, it is still possible to recognize the structure of the given statements and write a meaningful negation of that statement.

In abstract algebra, an operation $*$ on a set $A$ is called a commutative operation

provided that for all $x, y \in A$, $x * y = y * x$. Carefully explain what it means to say that an operation $*$ on a set $A$ is not a commutative operation.

The following exercises contain definitions or results from more advanced mathematics courses. Even though we may not understand all of the terms involved, it is still possible to recognize the structure of the given statements and write a meaningful negation of that statement.

In abstract algebra, a ring

consists of a nonempty set $R$ and two operations called addition and multiplication. A nonzero element $a$ in a ring $R$ is called a zero divisor

provided that there exists a nonzero element $b$ in $R$ such that $ab = 0$ or $ba = 0$. Carefully explain what it means to say that a nonzero element $a$ in a ring $R$ is not a zero divisor.