Chapter 1.8, Problem 10PSC

To determine

The conclusion from the following statements

Answer to Problem 10PSC

At least one of the given statements is false.

Explanation of Solution

Given information: We rewrite the statements in symbolic form.

  $\begin{array}{l} l\Rightarrow h ----Equation1\hfill \\ m\Rightarrow ~h ----Equation2\hfill \\ l\Rightarrow m ----Equation3\hfill \end{array}$

Concept Used:

Theorem: If a conditional statement is true, then the contrapositive of the statement is also true.

  $(Ifp,thenq\iff If~q,then~p.)$

In other words, a statement and contrapositive are logically equivalent.

Calculation:

  $(Ifp,thenq\iff If~q,then~p.)$

To complete the chain of reasoning, we will apply above theorem to Equation 1 and we get

  $~h\Rightarrow ~l$

  $\begin{array}{l}Ifwesubstitutem\Rightarrow hinaboveequation,weget\hfill \\ m\Rightarrow l\hfill \\ Theequationm \Rightarrow ~lisnotequivalenttoEquation3.\hfill \end{array}$

Conclusion:

It can be concluded that first and second statements are true while last statement is false.