Chapter 1.8, Problem 9PSB

To determine

The conclusion for the following statements.

Answer to Problem 9PSB

  $“Ifp,then~g”$

Explanation of Solution

Given information: The statements are as follows:

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

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:

  $\begin{array}{l}ItcanbeseenthatEquation1iscontrapositiveof f\Rightarrow c.\hfill \\ f \Rightarrow c ----Equation5\hfill \end{array}$

From Equation 4 and Equation 5, we get

  $f\Rightarrow ~b ----Equation6$

From Equation 3 and Equation 6, we get

  $p\Rightarrow ~b ----Equation7$

  $\begin{array}{l}ItcanbeseenthatEquation2isconverseofb\Rightarrow g.\hfill \\ b\Rightarrow g ----Equation8\hfill \end{array}$

From Equation 7 and Equation 8, we get

  $p\Rightarrow ~g$

Conclusion:

  $“Ifp,then~g”,i.e. p\Rightarrow ~g.$