The negation of the statement pattern $\sim \mathrm{S} \vee(\sim \mathrm{r} \wedge \mathrm{s})$ is…

The negation of the statement pattern $\sim \mathrm{S} \vee(\sim \mathrm{r} \wedge \mathrm{s})$ is equivalent to
  1. $\mathrm{s} \wedge \mathrm{r}$
  2. $\mathrm{s} \wedge(\mathrm{r} \wedge \sim \mathrm{s})$
  3. $\mathrm{s} \wedge \sim \mathrm{r}$
  4. $\mathrm{S} \vee(\mathrm{r} \vee \sim \mathrm{s})$

Solution

$\begin{aligned} & \sim(\sim s \vee(\sim r \wedge s)) \\ & \equiv s \wedge \sim(\sim r \wedge s)...[De Morgan's law] \\ & \equiv s \wedge(r \vee \sim s)......[De Morgan's law] \\ & \equiv(s \wedge r) \vee(s \wedge \sim s)...[Distributive law] \\ & \equiv(s \wedge r) \vee F...[Complement law] \\ & \equiv s \wedge r...[Identity law] \end{aligned}$

Asked in: MHT CET 2023 (10 May Shift 2)

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