Chapter B.1, Problem 5E

Solution

(a.)

The negation of the given statement.

It has been determined that the negation of the given statement is “The book does not have 500 pages.”

Given:

The book has 500 pages.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “The book has 500 pages.”

Then, the negation of the given statement is “The book does not have 500 pages.”

Conclusion:

It has been determined that the negation of the given statement is “The book does not have 500 pages.”

(b.)

The negation of the given statement.

It has been determined that the negation of the given statement is “Six is not less than eight.”

Given:

Six is less than eight.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Six is less than eight.”

Then, the negation of the given statement is “Six is not less than eight.”

Conclusion:

It has been determined that the negation of the given statement is “Six is not less than eight.”

(c.)

The negation of the given statement.

It has been determined that the negation of the given statement is “ $3\cdot 5\ne 15$ .”

Given:

  $3\cdot 5=15$

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “ $3\cdot 5=15$ .”

Then, the negation of the given statement is “ $3\cdot 5\ne 15$ .”

Conclusion:

It has been determined that the negation of the given statement is “ $3\cdot 5\ne 15$ .”

(d.)

The negation of the given statement.

It has been determined that the negation of the given statement is “No person has blonde hair.”

Given:

Some people have blonde hair.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Some people have blonde hair.”

Then, the negation of the given statement is “No person has blonde hair.”

Conclusion:

It has been determined that the negation of the given statement is “No person has blonde hair.”

(e.)

The negation of the given statement.

It has been determined that the negation of the given statement is “Some dogs do not have four legs.”

Given:

All dogs have four legs.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “All dogs have four legs.”

Then, the negation of the given statement is “Some dogs do not have four legs.”

Conclusion:

It has been determined that the negation of the given statement is “Some dogs do not have four legs.”

(f.)

The negation of the given statement.

It has been determined that the negation of the given statement is “All cats have nine lives.”

Given:

Some cats do not have nine lives.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Some cats do not have nine lives.”

Then, the negation of the given statement is “All cats have nine lives.”

Conclusion:

It has been determined that the negation of the given statement is “All cats have nine lives.”

(g.)

The negation of the given statement.

It has been determined that the negation of the given statement is “Some squares are not rectangles.”

Given:

All squares are rectangles.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “All squares are rectangles.”

Then, the negation of the given statement is “Some squares are not rectangles.”

Conclusion:

It has been determined that the negation of the given statement is “Some squares are not rectangles.”

(h.)

The negation of the given statement.

It has been determined that the negation of the given statement is “All rectangles are squares.”

Given:

Not all rectangles are squares.

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Not all rectangles are squares.”

Then, the negation of the given statement is “All rectangles are squares.”

Conclusion:

It has been determined that the negation of the given statement is “All rectangles are squares.”

(i.)

The negation of the given statement.

It has been determined that the negation of the given statement is “For some natural numbers $x$ , $x+3\ne 3+x$ .”

Given:

For all natural numbers $x$ , $x+3=3+x$ .

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “For all natural numbers $x$ , $x+3=3+x$ .”

Then, the negation of the given statement is “For some natural numbers $x$ , $x+3\ne 3+x$ .”

Conclusion:

It has been determined that the negation of the given statement is “For some natural numbers $x$ , $x+3\ne 3+x$ .”

(j.)

The negation of the given statement.

It has been determined that the negation of the given statement is “ $3\cdot \left(x+2\right)\ne 12$ for all natural numbers $x$ .”

Given:

There exists a natural number $x$ such that $3\cdot \left(x+2\right)=12$ .

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “There exists a natural number $x$ such that $3\cdot \left(x+2\right)=12$ .”

Then, the negation of the given statement is “ $3\cdot \left(x+2\right)\ne 12$ for all natural numbers $x$ .”

Conclusion:

It has been determined that the negation of the given statement is “ $3\cdot \left(x+2\right)\ne 12$ for all natural numbers $x$ .”

(k.)

The negation of the given statement.

It has been determined that the negation of the given statement is “Some counting numbers is not divisible by itself and $1$ .”

Given:

Every counting number is divisible by itself and $1$ .

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Every counting number is divisible by itself and $1$ .”

Then, the negation of the given statement is “Some counting numbers is not divisible by itself and $1$ .”

Conclusion:

It has been determined that the negation of the given statement is “Some counting numbers is not divisible by itself and $1$ .”

(l.)

The negation of the given statement.

It has been determined that the negation of the given statement is “All natural numbers are divisible by $2$ .”

Given:

Not all natural numbers are divisible by $2$ .

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “Not all natural numbers are divisible by $2$ .”

Then, the negation of the given statement is “All natural numbers are divisible by $2$ .”

Conclusion:

It has been determined that the negation of the given statement is “All natural numbers are divisible by $2$ .”

(m.)

The negation of the given statement.

It has been determined that the negation of the given statement is “For some natural numbers $x$ , $5x+4x\ne 9x$ .”

Given:

For all natural numbers $x$ , $5x+4x=9x$ .

Concept used:

The negation of a statement is a statement that has the opposite truth value compared to the original statement.

The negation of a statement with universal quantifier is a statement with existential quantifier and vice versa.

Calculation:

The given statement is “For all natural numbers $x$ , $5x+4x=9x$ .”

Then, the negation of the given statement is “For some natural numbers $x$ , $5x+4x\ne 9x$ .”

Conclusion:

It has been determined that the negation of the given statement is “For some natural numbers $x$ , $5x+4x\ne 9x$ .”