Ali; starting with the equality $x = y$ for non-zero, equal real numbers x and y, follows the following steps in order.
I. Let us multiply both sides of the equality by x: $$x ^ { 2 } = x \cdot y$$
II. Let us subtract $\mathrm { y } ^ { 2 }$ from both sides: $$x ^ { 2 } - y ^ { 2 } = x \cdot y - y ^ { 2 }$$
III. Let us factor both sides: $$( x + y ) ( x - y ) = y ( x - y )$$
IV. Let us divide both sides by $\mathrm { x } - \mathrm { y }$: $$x + y = y$$
V. Let us substitute y for x: $$2 y = y$$
As a result of these steps, Ali arrives at the conclusion "Every number equals twice itself."
Accordingly, in which of the numbered steps did Ali make an error?
A) I B) II C) III D) IV E) V

For propositions $p$, $q$, and $r$ $$( p \Rightarrow q ) \Rightarrow r$$ it is known that the proposition is false.
Accordingly,\ I. $p \Rightarrow q$\ II. $q \Rightarrow r$\ III. $r \Rightarrow p$\ Which of the following propositions are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and III\ E) II and III

A student made an error while proving the following claim that he believed to be true.
Claim: The number $\pi$ equals the number $e$.\ The student's proof: Let $f ( x )$ and $g ( x )$ be functions for $x > 0$ defined as $\mathrm{f} ( \mathrm{x} ) = \ln ( \pi \mathrm{x} )$ and $\mathrm{g} ( \mathrm{x} ) = \ln ( \mathrm{ex} )$.\ I. For every $x > 0$, the derivatives of functions $f ( x )$ and $g ( x )$ are equal to each other.\ II. Therefore, for every $x > 0$, functions $f ( x )$ and $g ( x )$ are equal to each other.\ III. Since $\ln ( x )$ is one-to-one and $f ( x ) = g ( x )$, we conclude that for every $x > 0$, $\pi x = ex$.\ IV. If two functions are equal for every $x > 0$, then their values at $x = 1$ are the same.\ V. Since the values of the functions $\pi \mathrm{x}$ and $ex$ at $x = 1$ are the same, we conclude that $\pi = \mathrm{e}$.\ In which of the numbered steps did this student make an error?\ A) I\ B) II\ C) III\ D) IV\ E) V

Let $a$ and $b$ be integers. The notation $\mathrm { a } \mid \mathrm { b }$ means that $a$ divides $b$ exactly.
A student wants to prove that the proposition "If integers $a$, $b$ and $c$ satisfy the conditions $a \mid c$ and $b \mid c$, then $(a + b) \mid c$ also holds." is false by using the counterexample method.
Accordingly, which of the following could be the example given by the student?