Question

Find a counterexample, if possible, to these universallyquantified statements, where the domain for all variables consists of all integers.
a)\(\forall \user1{x}\exists y\left( {\user1{x = }{\raise0.7ex\hbox{\(\user1{1}\)} \!\mathord{\left/

{\vphantom {\user1{1} \user1{y}}}\right.\kern-\nulldelimiterspace}

\!\lower0.7ex\hbox{\(\user1{y}\)}}\user1{ }} \right)\)
b)\(\forall \user1{x}\exists y\left( {{\user1{y}^\user1{2}}\user1{ - x < 100}} \right)\)
c)\(\forall \user1{x}\forall y\left( {{\user1{x}^\user1{2}} \ne {\user1{y}^\user1{2}}} \right)\)

Short answer

a) The Counter example for the given statement is,

For\(x = 0\)there is no y integer in such case,\(0 = \frac{1}{y}\)

b) The Counter example for the given statement is,

For \(x = - 100\)there is no y integer in such case, \({y^2} + 100 < 100\)

c) The Counter example for the given statement is,

\(x = 8{\rm{ }}and{\rm{ y}} = 4{\rm{,for }}{x^2} = {y^3} = 64\).

Step-by-step solution

Introduction to the Concept

Negation: A statement's negation is the inverse of the provided mathematical statement.

Quantifiers: A quantifier is a term that normally precedes a noun and expresses the object's amount.

Solution Explanation

a)

Here the given statement is,

\(\forall \user1{x}\exists y\left( {\user1{x = }{\raise0.7ex\hbox{$\user1{1}$} \!\mathord{\left/

{\vphantom {\user1{1} \user1{y}}}\right.\kern-\nulldelimiterspace}

\!\lower0.7ex\hbox{$\user1{y}$}}\user1{ }} \right)\)

Let’s determine thatthe given statement is True or False,

Nowthe universally quantified statement is false.

Considering all x, y is there in such a way, the statement is given as \(x = \frac{1}{y}\)yet for\(x = 0\)and there is no y integer in such case, \(x = \frac{1}{y}\).

Solution Explanation

b)

Here the given statement is,

\(\forall \user1{x}\exists y\left( {{\user1{y}^\user1{2}}\user1{ - x < 100}} \right)\)

Let’s determine that the given statement is True or False,

Nowthe universally quantified statement is false.

Considering all x, y is there in such a way, the statement is given as \({y^2} - x < 100\)yet for\(x = - 100\)

\(\begin{ALIGNED}{l}{y^2} - \left( { - 100} \right) < 100\\ \Rightarrow {y^2} < 0\end{ALIGNED}\)

There is no possible y integer.

Solution Explanation

c)

Here the given statement is,

\(\forall \user1{x}\forall y\left( {{\user1{x}^\user1{2}} \ne {\user1{y}^\user1{2}}} \right)\)

Let’s determine thatthe given statement is True or False,

Nowthe universally quantified statement is false.

Considering all x and y, the statement is given as\({x^2} \ne {y^3}\)yet for

\(x = 8{\rm{ }}and{\rm{ y}} = 4{\rm{,for }}{x^2} = {y^3} = 64\).