Question

"If A is true, then B is false", is logically equivalent to which of the following?
I. If A is false, then B is true.
II. If B is false, then A is true.
III. If B is true, then A is false
$A)$None of the above
$B)$III only
$C)$I and II only
$D)$II and III only
$E)$I, II, and III

Answer 1

First we have to define what the terms we need to solve the problem are.
Since the given question is If A is true then B is needed to be false and we need to find the equivalent form of the expression from the I, II, III statements. Equivalent means more likely similar expressions but not the same. So, we need to find a more similar statement.

Complete step-by-step solution:
Since the given known statement is if A is true then B needs to be false from this, we need to find the required answer using the I, II, III statements. Starting with the statement I which is if A is false then B is true; this expression is not possible because from the given statement they clearly say that A is not logically replicated to B thus it is wrong. Hence options $C)$I and II only and $E)$I, II, and III are eliminated (containing statements I). next check if B is false then A needs to be true is again not possible because from the given statements B is not repleted to A and hence it is false $D)$II and III only eliminated
Similar to the same method we can say III. If B is true, then A is false; which is B is true and then A is true or A is false, which is the only statement similar to the given statement and hence $B)$ III only is the correct option

Note: Since option, $A)$None of the above means no options are correct, but here we get the correct option B and hence option A is eliminated. The only statement equivalent to If A is true, then B is false is If B is true, then A is false.