Chapter B.2, Problem 12E

Solution

(a.)

To Translate: The given statement into symbolic form, with the meaning of the symbols used.

It has been determined that the given statement can be written symbolically as $p\to \left(q\wedge r\right)$ , where $p$ denotes “Mary’s little lamb follows her to school”, $q$ denotes “the appearance of Mary’s little lamb in the school will break the rules” and $r$ denotes “Mary will be sent home”.

Given:

If Mary’s little lamb follows her to school, then its appearance there will break the rules and Mary will be sent home.

Concept used:

  $p\to q$ means if $p$ , then $q$ .

  $p\wedge q$ means $p$ and $q$ .

  $p\vee q$ means $p$ or $q$ .

Calculation:

Let $p$ denote “Mary’s little lamb follows her to school”, $q$ denote “the appearance of Mary’s little lamb in the school will break the rules” and $r$ denote “Mary will be sent home”.

The given statement is:

If Mary’s little lamb follows her to school, then its appearance there will break the rules and Mary will be sent home.

Symbolically, the above statement can be written as:

If $p$ , then $q$ and $r$ .

On simplification,

If $p$ , then $q\wedge r$ .

On further simplification,

  $p\to \left(q\wedge r\right)$

Conclusion:

It has been determined that the given statement can be written symbolically as $p\to \left(q\wedge r\right)$ , where $p$ denotes “Mary’s little lamb follows her to school”, $q$ denotes “the appearance of Mary’s little lamb in the school will break the rules” and $r$ denotes “Mary will be sent home”.

(b.)

To Translate: The given statement into symbolic form, with the meaning of the symbols used.

It has been determined that the given statement can be written symbolically as $\sim \left(p\wedge q\right)\to \sim r$ , where $p$ denotes “Jack is nimble”, $q$ denotes “Jack is quick” and $r$ denotes “Jack will make it over the candlestick”.

Given:

If it is not the case that Jack is nimble and quick, then Jack will not make it over the candlestick.

Concept used:

  $\sim p$ means not $p$ .

  $p\to q$ means if $p$ , then $q$ .

  $p\wedge q$ means $p$ and $q$ .

  $p\vee q$ means $p$ or $q$ .

Calculation:

Let $p$ denote “Jack is nimble”, $q$ denote “Jack is quick” and $r$ denote “Jack will make it over the candlestick”.

The given statement is:

If it is not the case that Jack is nimble and quick, then Jack will not make it over the candlestick.

Symbolically, the above statement can be written as:

If it is not the case that $p$ and $q$ , then not $r$ .

On simplification,

If it is not the case that $p\wedge q$ , then $\sim r$ .

On further simplification,

If $\sim \left(p\wedge q\right)$ , then $\sim r$ .

Finally,

  $\sim \left(p\wedge q\right)\to \sim r$

Conclusion:

It has been determined that the given statement can be written symbolically as $\sim \left(p\wedge q\right)\to \sim r$ , where $p$ denotes “Jack is nimble”, $q$ denotes “Jack is quick” and $r$ denotes “Jack will make it over the candlestick”.

(c.)

To Translate: The given statement into symbolic form, with the meaning of the symbols used.

It has been determined that the given statement can be written symbolically as $\sim p\to \sim q$ , where $p$ denotes “the apple had hit Isaac Newton on the head” and $q$ denotes “the laws of gravity would have been discovered”.

Given:

If the apple had not hit Isaac Newton on the head, then the laws of gravity would not have been discovered.

Concept used:

  $\sim p$ means not $p$ .

  $p\to q$ means if $p$ , then $q$ .

  $p\wedge q$ means $p$ and $q$ .

  $p\vee q$ means $p$ or $q$ .

Calculation:

Let $p$ denote “the apple had hit Isaac Newton on the head” and $q$ denote “the laws of gravity would have been discovered”.

The given statement is:

If the apple had not hit Isaac Newton on the head, then the laws of gravity would not have been discovered.

Symbolically, the above statement can be written as:

If not $p$ , then not $q$ .

On simplification,

If $\sim p$ , then $\sim q$ .

Finally,

  $\sim p\to \sim q$

Conclusion:

It has been determined that the given statement can be written symbolically as $\sim p\to \sim q$ , where $p$ denotes “the apple had hit Isaac Newton on the head” and $q$ denotes “the laws of gravity would have been discovered”.