Chapter B.1, Problem 9E

Solution

(a.)

A set that corresponds by analogy to the given statement, using set operations.

It has been determined that a set that corresponds by analogy to the given statement is $R\cup S$ .

Given:

  $r\vee s$

Concept used:

The logical symbols $\wedge$ , $\vee$ and $\sim$ corresponds to the set operations: set intersection ( $\cap$ ), set union ( $\cup$ ) and set complement ( ${}^C$ ) respectively.

Calculation:

The given statement is $r\vee s$ .

As discussed, $\vee$ corresponds to set union ( $\cup$ ).

Then, a set that corresponds by analogy to the given statement is $R\cup S$ .

Conclusion:

It has been determined that a set that corresponds by analogy to the given statement is $R\cup S$ .

(b.)

A set that corresponds by analogy to the given statement, using set operations.

It has been determined that a set that corresponds by analogy to the given statement is $Q\cap Q^C$ .

Given:

  $q\wedge \sim q$

Concept used:

The logical symbols $\wedge$ , $\vee$ and $\sim$ corresponds to the set operations: set intersection ( $\cap$ ), set union ( $\cup$ ) and set complement ( ${}^C$ ) respectively.

Calculation:

The given statement is $q\wedge \sim q$ .

As discussed, $\wedge$ corresponds to set intersection ( $\cap$ ) and $\sim$ , to set complement ( ${}^C$ ).

Then, a set that corresponds by analogy to the given statement is $Q\cap Q^C$ .

Conclusion:

It has been determined that a set that corresponds by analogy to the given statement is $Q\cap Q^C$ .

(c.)

A set that corresponds by analogy to the given statement, using set operations.

It has been determined that a set that corresponds by analogy to the given statement is ${\left(R\cup Q\right)}^C$ .

Given:

  $\sim \left(r\vee q\right)$

Concept used:

The logical symbols $\wedge$ , $\vee$ and $\sim$ corresponds to the set operations: set intersection ( $\cap$ ), set union ( $\cup$ ) and set complement ( ${}^C$ ) respectively.

Calculation:

The given statement is $\sim \left(r\vee q\right)$ .

As discussed, $\vee$ corresponds to set union ( $\cup$ ) and $\sim$ , to set complement ( ${}^C$ ).

Then, a set that corresponds by analogy to the given statement is ${\left(R\cup Q\right)}^C$ .

Conclusion:

It has been determined that a set that corresponds by analogy to the given statement is ${\left(R\cup Q\right)}^C$ .

(d.)

A set that corresponds by analogy to the given statement, using set operations.

It has been determined that a set that corresponds by analogy to the given statement is $P\cap \left(R\cup S\right)$ .

Given:

  $p\wedge \left(r\vee s\right)$

Concept used:

The logical symbols $\wedge$ , $\vee$ and $\sim$ corresponds to the set operations: set intersection ( $\cap$ ), set union ( $\cup$ ) and set complement ( ${}^C$ ) respectively.

Calculation:

The given statement is $p\wedge \left(r\vee s\right)$ .

As discussed, $\vee$ corresponds to set union ( $\cup$ ) and $\wedge$ , to set intersection ( $\cap$ ).

Then, a set that corresponds by analogy to the given statement is $P\cap \left(R\cup S\right)$ .

Conclusion:

It has been determined that a set that corresponds by analogy to the given statement is $P\cap \left(R\cup S\right)$ .