Chapter 1.8, Problem 5PSA

a.

To determine

The conclusion for the following statements.

Answer to Problem 5PSA

“If d, then f”

Explanation of Solution

Given information: The statements are as follows:

  $\begin{array}{l}a\Rightarrow b ----Equation1\hfill \\ d\Rightarrow ~c ----Equation2\hfill \\ ~c\Rightarrow a ----Equation3\hfill \\ b\Rightarrow f ----Equation4\hfill \end{array}$

Concept Used:

Every conditional statement “If p, then q” has three other statements associated with it.

A converse (If q, then p)

An inverse $(If~p,then~q.)$

A contrapositive $(If~q,then~p.)$

Calculation:

  $\begin{array}{l}FromEquation1andEquation4,weget\hfill \\ a\Rightarrow f ----Equation5\hfill \\ FromEquation2andEquation3,weget\hfill \\ d\Rightarrow a ----Equation6\hfill \\ FromEquation5andEquation6,weget\hfill \\ d\Rightarrow f \hfill \end{array}$

Conclusion:

  $"Ifd,thenf",i.e.d\Rightarrow f.$

b.

To determine

The conclusion for the given statements.

Answer to Problem 5PSA

“If s, then (p”

Explanation of Solution

Given information: The statements are as follows:

  $\begin{array}{l}p\Rightarrow ~q ----Equation1\hfill \\ r\Rightarrow q ----Equation2\hfill \\ s\Rightarrow r ----Equation3\hfill \end{array}$

Concept Used:

Every conditional statement “If p, then q” has three other statements associated with it.

A converse (If q, then p)

An inverse $(If~p,then~q.)$

A contrapositive $(If~q,then~p.)$

Calculation:

  $\begin{array}{l}IwetakeconverseofEquation1,weget\hfill \\ q\Rightarrow ~p ----Equation4\hfill \\ FromEquation3andEquation2,weget\hfill \\ s\Rightarrow q ----Equation5\hfill \\ FromEquation4andEquation5,weget\hfill \\ s\Rightarrow ~p\hfill \end{array}$

Conclusion:

  $"Ifs,then~p",i.e.s\Rightarrow ~p.$

c.

To determine

To conclude a result from the statements.

Answer to Problem 5PSA

If bobcats begin to browse, then horses head for home.

Explanation of Solution

Given information: We rewrite the statements in symbolic form. $\begin{array}{l}w\Rightarrow c ----Equation1\hfill \\ g\Rightarrow h ----Equation2\hfill \\ c\Rightarrow g ----Equation3\hfill \\ b\Rightarrow w ----Equation4\hfill \end{array}$

Concept Used:

Every conditional statement “If p, then q” has three other statements associated with it.

A converse (If q, then p)

An inverse $(If~p,then~q.)$

A contrapositive $(If~q,then~p.)$

Calculation:

From Equation 4 and Equation 1, we get

  $b\Rightarrow c ----Equation5$

From Equation 3 and Equation 2, we get

  $c\Rightarrow h ----Equation6$

From Equation 5 and Equation 6, we get

  $b\Rightarrow h$Conclusion:

  $"Ifb,thenh",i.e.b\Rightarrow h.$