Here’s the breakdown and solutions for each part of the question:

(a) The real sequence $(x_n)$ is not convergent if...

This can be expressed using quantifiers: $\neg (\exists L \in \mathbb{R})(\forall \epsilon > 0)(\exists N \in \mathbb{N})(\forall n \geq N, |x_n - L| < \epsilon).$ This negation essentially states that there is no real number $L$ such that $(x_n)$ satisfies the definition of convergence.

(b) Write the contrapositive of "For real numbers $x, y \in \mathbb{R}$, if $xy = 0$, then $x = 0$ or $y = 0$."

The contrapositive of $P \implies Q$ is $\neg Q \implies \neg P$.

• Original Statement: $xy = 0 \implies (x = 0 \lor y = 0).$
• Contrapositive: $\neg (x = 0 \lor y = 0) \implies \neg (xy = 0).$

Simplify:

• $\neg (x = 0 \lor y = 0) = (x \neq 0 \land y \neq 0).$
• $\neg (xy = 0) = xy \neq 0.$

Thus, the contrapositive is: $(x \neq 0 \land y \neq 0) \implies xy \neq 0.$

(c) Write the negation of the statement "((\forall x, y \in \mathbb{R})(x < y \implies x^2 < y^2)."

To negate a universal quantifier, replace it with an existential quantifier and negate the inner statement.

Original: $(\forall x, y \in \mathbb{R})(x < y \implies x^2 < y^2).$

Negation: $(\exists x, y \in \mathbb{R})(x < y \land x^2 \geq y^2).$

(d) Can the set $F = \{(x, y) : x^3 + y = 1\} \subseteq \mathbb{R}^2$ be the graph of a function from a subset of $\mathbb{R}$ to $\mathbb{R}$? Justify.

To check if $F$ can represent the graph of a function, each $x \in \mathbb{R}$ must map to at most one $y$.

• The equation $x^3 + y = 1$ implies $y = 1 - x^3$, which is uniquely determined for each $x \in \mathbb{R}$.
• Hence, $F$ is the graph of the function $f(x) = 1 - x^3$.

Let me know if you'd like further clarification or explanations!

Related Questions:

1. What is the definition of convergence for a real sequence?
2. How do we determine the negation of a universally quantified statement?
3. What are examples of other functions whose graphs are determined by unique mappings like $y = 1 - x^3$?
4. Why does the contrapositive of a statement logically hold true if the original statement is true?
5. What happens if $x^3 + y = 1$ is modified to $x^2 + y^2 = 1$? Can it still represent a function?

Tip:

When dealing with logical quantifiers, always break the problem into smaller steps: identify the structure (universal/existential), apply negation systematically, and simplify.