Chapter 1.3, Problem 1E

Translate the following English sentences into symbolic sentences with quantifiers. The universe for each is given in parentheses.

1. Not all precious stones are beautiful. (All stones)
2. All precious stones are not beautiful. (All stones)
3. Some isosceles triangle is a right triangle. (All triangles)
4. No right triangle is isosceles. (All triangles)
5. Every triangle that is not isosceles is a right triangle.
6. All people are honest or no one is honest. (All people)
7. Some people are honest and some people are not honest. (All people)
8. Every nonzero real number is positive or negative. (Real numbers)
9. Every integer is greater than -4 or less than 6. (Real numbers)
10. Every integer is greater than some integer. (Integers)
11. No integer is greater than every other integer. (Integers)
12. Between any integer and any larger integer, there is a real number. (Real numbers)
13. There is a smallest positive integer. (Real numbers)
14. No one loves everybody. (All people)
15. Everybody loves someone. (All people)
16. For every positive real number x, there is a unique real number y such that $2^y=x.$ (Real numbers)

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is not beautiful).

Explanation of Solution

Given:

It is given in the question that Not all the precious stones are beautiful (All stones).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceNot all the precious stones are beautiful (All stones) can be written in symbolic quantifiers as $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is not beautiful).

Conclusion:

  $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is not beautiful).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is beautiful).

Explanation of Solution

Given:

It is given in the question that All precious stone are not beautiful.(all stones).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceAll precious stone are not beautiful.(all stones) can be written in symbolic quantifiers as $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is beautiful).

Conclusion:

  $(\exists x)$ ( $x$ is precious $\wedge$ $x$ is beautiful).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists x)$ ( $x$ is isosceles $\wedge$ $x$ is right angled).

Explanation of Solution

Given:

It is given in the question that Some isosceles triangle is a right triangle.(All triangles).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceSome isosceles triangle is a right triangle.(All triangles) can be written in symbolic quantifiers as $(\exists x)$ ( $x$ is isosceles $\wedge$ $x$ is right angled).

Conclusion:

  $(\exists x)$ ( $x$ is isosceles $\wedge$ $x$ is right angled).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ ( $x$ is right $\Rightarrow$ $x$ is not isosceles).

Explanation of Solution

Given:

It is given in the question that No right triangle is isosceles.(All triangles).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceNo right triangle is isosceles.(All triangles) can be written in symbolic quantifiers as $(\forall x)$ ( $x$ is right $\Rightarrow$ $x$ is not isosceles).

Conclusion:

  $(\forall x)$ ( $x$ is right $\Rightarrow$ $x$ is not isosceles).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ ( $x$ is not isosceles $\Rightarrow$ $x$ is right).

Explanation of Solution

Given:

It is given in the question that Every triangle that is not isosceles is a right triangle.

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceEvery triangle that is not isosceles is a right triangle can be written in symbolic quantifiers as $(\forall x)$ ( $x$ is not isosceles $\Rightarrow$ $x$ is right).

Conclusion:

  $(\forall x)$ ( $x$ is not isosceles $\Rightarrow$ $x$ is right).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ (x is honest) $\vee$ $(\forall x)$ (x is not honest).

Explanation of Solution

Given:

It is given in the question that All people are honest or no one is honest.(All people).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceAll people are honest or no one is honest.(All people) can be written in symbolic quantifiers as $(\forall x)$ ( x is honest) $\vee$

  $(\forall x)$ ( x is not honest).

Conclusion:

  $(\forall x)$ (x is honest) $\vee$ $(\forall x)$ (x is not honest).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ (x is honest $\vee$ x is not honest).

Explanation of Solution

Given:

It is given in the question that Some people are honest and some people are not honest (All people).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceSome people are honest and some people are not honest (All people) can be written in symbolic quantifiers as $(\forall x)$ ( x is honest $\vee$ x is not honest).

Conclusion:

  $(\forall x)$ (x is honest $\vee$ x is not honest).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ ( x is non zero $\wedge$ x is real) $\Rightarrow$ (x is positive) $\vee$ (x is negative).

Explanation of Solution

Given:

It is given in the question thatEvery nonzero real number is positive or negative.(Real numbers).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceEvery nonzero real number is positive or negative.(Real numbers) can be written in symbolic quantifiers as $(\forall x)$ ( x is non zero $\wedge$ x is real) $\Rightarrow$ ( x is positive) $\vee$ ( x is negative).

Conclusion:

  $(\forall x)$ ( x is non zero $\wedge$ x is real) $\Rightarrow$ (x is positive) $\vee$ (x is negative)

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ ( x is integer ) $\Rightarrow$ ( $x>-4$ $\vee$ $x<6$).

Explanation of Solution

Given:

It is given in the question that Every integer is greater than $-4$ or less than $6$ (Real numbers).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:$(\forall x)$ ( x is integer ) $\Rightarrow$ ( $x>-4$ $\vee$ $x<6$ )

Conclusion:

  $(\forall x)$ ( x is integer ) $\Rightarrow$ ( $x>-4$ $\vee$ $x<6$ )

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\forall x)$ ( xis integer) $\Rightarrow$ $(\forall y)$ ( y is integer $\wedge$ $y>x$ ).

Explanation of Solution

Given:

It is given in the question that Every integer is greater than some integer (Integers)

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceEvery integer is greater than some integer.(Integers) can be written in symbolic quantifiers as $(\forall x)$ ( x is integer) $\Rightarrow$

  $(\forall y)$ ( y is integer $\wedge$ $y>x$ ).

Conclusion:

  $(\forall x)$ ( x is integer) $\Rightarrow$ $(\forall y)$ ( y is integer $\wedge$ $y>x$ )

To determine

Translate the English sentence into symbolic sentence with quantifiers:

Answer to Problem 1E

The symbolic quantifiers of the sentence is $𑨄(\exists x)$ $(x\in Z)$ $\Rightarrow$ $(\forall y)$ ( y is integer $\wedge$ $y>x$ ).

Explanation of Solution

Given:

It is given in the question that No integer is greater than every other integer (Integers).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceNo integer is greater than every other integer (Integers) can be written in symbolic quantifiers as $𑨄(\exists x)$ $(x\in Z)$ $\Rightarrow$ $(\forall y)$ ( y is integer $\wedge$ $y>x$ ).

Conclusion:

  $𑨄(\exists x)$ $(x\in Z)$ $\Rightarrow$ $(\forall y)$ ( y is integer $\wedge$ $y>x$ )

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists x)$ $(\forall y)$ $(\forall z)$ $(y\in Z\wedge z\in Z\wedge z>y)$ $\Rightarrow$ ( $x>y$ $\wedge$ $x<z$ ).

Explanation of Solution

Given:

It is given in the question that Between any integer and any larger integer,there is a real number.(Real numbers).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceBetween any integer and any larger integer,there is a real number.(Real numbers) can be written in symbolic quantifiers as $(\exists x)$ $(\forall y)$ $(\forall z)$ $(y\in Z\wedge z\in Z\wedge z>y)$ $\Rightarrow$ ( $x>y$ $\wedge$ $x<z$ ).

Conclusion:

  $(\exists x)$ $(\forall y)$ $(\forall z)$ $(y\in Z\wedge z\in Z\wedge z>y)$ $\Rightarrow$ ( $x>y$ $\wedge$ $x<z$ ).

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists x)$ (( $x\in Z$ ) $\wedge$ $x>0$ $\wedge$ $x<y$ $\forall$ $y\in Z$ )).

Explanation of Solution

Given:

It is given in the question that There is a smallest positive integer (Real numbers).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentencethere is a smallest positive integer.(Real numbers) can be wriiten in symbolic quantifiers as $(\exists x)$ (( $x\in Z$ ) $\wedge$ $x>0$ $\wedge$ $x<y$ $\forall$ $y\in Z$ )).

Conclusion:

  $(\exists x)$ (( $x\in Z$ ) $\wedge$ $x>0$ $\wedge$ $x<y$ $\forall$ $y\in Z$ ))

To determine

Translate the English sentence into symbolic sentence with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $𑨄(\exists x)(\forall y)$ ( x loves y $\wedge$ $x\ne y$ ).

Explanation of Solution

Given:

It is given in the question that No one loves everybody.(All people).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation: The sentenceNo one loves everybody.(All people) can be written in symbolic quantifiers as

  $𑨄(\exists x)(\forall y)$ ( x loves y $\wedge$ $x\ne y$ ).

Conclusion:

  $𑨄(\exists x)(\forall y)$ ( x loves y $\wedge$ $x\ne y$ )

To determine

Translate the English sentence into symbolic sentences with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists y)(\forall x)$ ( x loves y ).

Explanation of Solution

Given:

It is given in the question that Everybody loves someone (All People).

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceEverybody loves someone (All People) can be wriiten in symbolic quantifiers as $(\exists y)(\forall x)$ ( x loves y ).

Conclusion:

  $(\exists y)(\forall x)$ ( x loves y )

To determine

Translate the English sentence into symbolic sentences with quantifiers.

Answer to Problem 1E

The symbolic quantifiers of the sentence is $(\exists y)(\forall x)$ ( $x>0$ $\wedge$ $2^y=x$ ).

Explanation of Solution

Given:

It is given in the question that For every positive real number x , there is a unique real number y such that $2^y=x$ (Real numbers)

Concept Used:

In this we have to use the concept of logic and proofs.

Calculation:

The sentenceFor every positive real number x , there is a unique real number y such that $2^y=x$ (Real numbers) can be written in symbolic quantifiers as $(\exists y)(\forall x)$ ( $x>0$ $\wedge$ $2^y=x$ )

Conclusion: $(\exists y)(\forall x)$ ( $x>0$ $\wedge$ $2^y=x$ )