Transitive Property of Implication

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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?

- $S$ is a sufficient condition for $R$.
- $S$ is a necessary condition for $R$.
- $S$ is neither sufficient, nor a necessary condition for $R.$
- $S$ is a sufficient and necessary condition for $R$.

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 $P$.

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 $P$.