Hello Khonics89 Originally Posted by

**Khonics89** For each real number a consider the proposition p(a) given by

∃x ∀y (square root of|x| > and = a|y|) ⇒ (|x| > and = y^2),

where the universal set U is the set of real numbers.

(i ) Apply the negation rule to the quantifiers to write ∼p(a) without using the implication

sign ⇒.

(ii ) Find all values of a for which p(a) is true. Justify your answer.

A little confused as to what the answer is, because there is no answers in the back.... :\

Can you show me your way and if you can also include working out thanks

Two things to note first:

- When we negate quantifiers, we flip all $\displaystyle \forall$ and $\displaystyle \exists$ signs and negate any propositions.

- The proposition $\displaystyle p \Rightarrow q$, written without a $\displaystyle \Rightarrow$ sign, is equivalent to $\displaystyle \neg p \lor q$.

So $\displaystyle \sim\Big(\exists x\, \forall y\,[p \Rightarrow q]\Big)$ $\displaystyle \equiv\, \sim\Big(\exists x\, \forall y\,[\neg p \lor q]\Big)$

$\displaystyle \equiv \forall x\, \exists y\, [p \land \neg q]$, using De Morgan's Law

(i) So if $\displaystyle p(a)\equiv \exists x\, \forall y\,[(\sqrt{|x|}\ge a|y|) \Rightarrow (|x| \ge y^2)]$, then$\displaystyle \sim p(a)\equiv \forall x\, \exists y\, [(\sqrt{|x|}\ge a|y|) \land (|x| < y^2)]$

(ii) Looking first at $\displaystyle \sim p(a)$, then, we have to find the values of $\displaystyle a$ for which the following statement is true:Given an $\displaystyle x$, we can find a $\displaystyle y$ such that

- $\displaystyle \sqrt{|x|}\ge a|y|$; and

- $\displaystyle |x| < y^2$

So what sort of value will $\displaystyle a$ have?

First, consider $\displaystyle a\le 0$.

$\displaystyle a \le 0 \Rightarrow a|y| \le 0$ for any value of $\displaystyle y$, and hence $\displaystyle \sqrt{|x|} \ge a|y|$ will be true for any $\displaystyle x$ and any $\displaystyle y$. So provided we choose $\displaystyle y$ such that $\displaystyle |x| <y^2$, both statements will be true.

Hence $\displaystyle \sim p(a)$ is true for any $\displaystyle a \le 0$.

Now let's consider $\displaystyle a >0$.

In this case both sides of the inequality $\displaystyle \sqrt{|x|}\ge a|y|$ are non-negative for all values of $\displaystyle x$ and $\displaystyle y$, and hence we may square both sides to get:

$\displaystyle |x| \ge a^2y^2$

In addition to this, we must also have $\displaystyle |x| < y^2$. Hence we can write:

$\displaystyle a^2y^2 \le |x| < y^2$

$\displaystyle \Rightarrow a^2y^2<y^2$

$\displaystyle \Rightarrow a^2<1$

$\displaystyle \Rightarrow |a| <1$

So for positive values of $\displaystyle a$, $\displaystyle \sim p(a)$ is true whenever $\displaystyle a<1$ .

Therefore $\displaystyle \sim p(a)$ is true for all $\displaystyle a < 1$, and hence $\displaystyle p(a)$ is true for $\displaystyle a \ge 1$.

Grandad