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Consider the following two statements.

  • $S_1$: If a candidate is known to be corrupt, then he will not be elected
  • $S_2$: If a candidate is kind, he will be elected


Which one of the following statements follows from $S_1$ and $S_2$ as per sound inference rules of logic?
 

  1. If a person is known to be corrupt, he is kind
  2. If a person is not known to be corrupt, he is not kind
  3. If a person is kind, he is not known to be corrupt
  4. If a person is not kind, he is not known to be corrupt
in Mathematical Logic by Veteran (105k points)
edited by | 2.6k views
0
Option C is correct.

S1:  P-->Q

S2:  X-->Z

If X is true then Z has to be true in order for statement S2 to be true.

If Z is true then Q is false. Now, if Q is false then P has to be false in order for statement S1 to be true.

4 Answers

+23 votes
Best answer

$\begin{align*} S_1 &= C \rightarrow \neg E\\ S_2 &= K \rightarrow E\\ \end{align*}$

so, writing them using primary operators :
$\begin{align*} S_1 &= \neg C \vee \neg E\\ S_2 &= \neg K \vee E\\ \end{align*}$

on using resolution principle
$\neg E$ and $E$ cancels each other out
and conclusion = $\neg C \vee \neg K$

which can also be written as $K \rightarrow \neg C$ which is translated into English as = option C

by Boss (30.8k points)
edited by
0
@amarVashishth

How do we apply resolution principle ?
+2
@junaid

If you have disjunction of literals given in the premises you can cut out the literals which are in true as well as complemented form in two different premises.
+51 votes

Option c. If a person is kind, he is not known to be corrupt

Let

$C(x): x \text{ is known to be corrupt}$
$K(x): x \text{ is kind}$
$E(x): x \text{ will be elected}$

  • $S1: C(x) \to \neg E(x)$
  • $S2: K(x) \to E(x)$

S1 can be written as $E(x) \to \neg C(x)$ as $A \to B = \neg B \to \neg A$.
Thus, from S1 and S2,

$K(x) \to E(x) \to \neg C(x)$.

Thus we get C option.

by Active (5k points)
+1
@anoop  
best explanation
+1
@anoop Best answer
0
Very good
+7 votes

Method for these kinds of question.

Use inference law : Here we use

Contrapositive law i.e. A $\rightarrow$ B is true then ~B $\rightarrow$ ~A is always true.

For  A $\leftrightarrow$ B is true the contrapositive true. converse(B$\leftrightarrow$ A) true and Inverse(~A$\leftrightarrow$~B)  also true.

by Veteran (63k points)
edited by
+2 votes

Let ,

        K: Person is kind

        C: Person is corrupt

        E: Person is elect

Here both statements 1,2 are premises and we need to check what is the conclusion.

S1: C-->¬E

S2: K-->E

3.  E-->¬C  from S1, and Contrapositive rule.

4. K-->¬C  from S2,3 and Hypothetical syllogism.It is valid.

K-->¬C ≡ " If a person is kind, he is not known to be corrupt ".

So, (c) is the Ans.

by Loyal (8k points)
Answer:

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