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I know $P \implies Q,$ I need to prove that $(P\implies Q)$ implies that $(R\implies S)$

Should I assume $R$ is true? [And $R$ is not the same as $P,$ or $Q$]

cos90
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  • Related – Mark Jan 06 '18 at 22:29
  • There are at least 5 ways to prove $A \implies B$: (1) Assume $A$ is true and prove $B$, (2) assume $B$ is false and prove $A$ is false, (3) prove both $A$ and $B$ are true, (4) prove we cannot have both $A$ is true and $B$ is false, or (5) prove $A$ is false. – Dan Christensen Jan 07 '18 at 23:15
  • Thanks for the clarification - this does indeed sense from the truth tables. – cos90 Jan 17 '18 at 02:36

2 Answers2

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You first assume $P \implies Q$ is true. Now you want to show $R \implies S$ is true. For this, you further assume $R$ is true and use $P \implies Q$ to show that $S$ is true.

For example, let

$P: A$ is the mother of $B$

$Q: C$ is a sister of $B$

$R: B$ is a son of $A$

$S: A$ is the mother of $C$

We want to show $(P \implies Q)\implies(R \implies S)$

Suppose, $P\implies Q$ and $R$ is true. So, we have that $B$ is a son of $A.$ This means that $A$ is a mother of $B.$ Since, $P \implies Q$ this means that $C$ is a sister of $B.$ Hence, $A$ must be the mother of $C.$

Sahiba Arora
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In order to show that $R \implies S$, you assume $R$ is true and prove $S$ is true. In the process you may or may not use $P \implies Q$.

Mohammad Riazi-Kermani
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