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Let $P,S,R$ be three statements(propositions). Let $S$ be a sufficient condition for $P$, Let $R$ is a necessary condition for $P$ then which of the following is/are true?
  1. $S$ is a sufficient condition for $R$.
  2. $S$ is a necessary condition for $R$.
  3. $S$ is neither sufficient, nor a necessary condition for $R.$
  4. $S$ is a sufficient and necessary condition for $R$.
in Mathematical Logic recategorized by
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Transitive Property of Implication
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3 Answers

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If $S → P$ and $P → R$ , then $S → R$.

This is called transitivity property or hypothetical syllogism. 

If $S → R$ then $S$ is a sufficient condition of $R$ and $R$ is a necessary condition for $S$.

Therefore $S$ is sufficient condition for $R$.

Answer $A$.

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Very clear explanation.
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2 votes
2 votes
$S$ is a sufficient condition for $P$, and $R$ is a necessary condition for $P$.
So, whenever $S$ is true, then $P$ is true. And whenever $P$ is true then $R$ is true. So, whenever $S$ is true, then $R$ is true(because when $S$ becomes true, then $P$ becomes true, then $R$ becomes true because $P$ has become true).
So, $S$ implies $R$. So, $S$ is a sufficient condition for $R$.
So, Whenever $S$ is true, $R$ is True.
So, sufficient condition for $P$ implies necessary condition for $P$.

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when p is true, how is r true? r is a necessary condition for p right? so how does r depend on p? can you please explain?
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If p is true then r has to be true because p->r has to be true as given in the question

if p is true and if r will be false then overall whole p→ r will be false.
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1 vote
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according to statement

s->p

p->R

then , s->p->R

so  option A is correct .

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Hypothetical Syllogism

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Answer:

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