The negation of the statement "If $5 2$, then $5>2 "$ is

The negation of the statement "If $5 < 7$ and $7>2$, then $5>2 "$ is
  1. $5 < 7$ and $7>2$ and $5 \leq 2$
  2. $5 < 7$ and $7>2$ or $5 < 2$
  3. $5 < 7$ and $7>2$ and $5>2$
  4. $5 < 7$ and $7>2$ or $5 \leq 2$

Solution

Let $\mathrm{p}: 5 < 7$ and $\mathrm{q}: 7>2$ and $\mathrm{r}: 5>2 .$ The logical form of given statement is $(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{r}$ $\therefore \quad[(\mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{r}] \equiv \sim[\sim(\mathrm{p} \wedge \mathrm{q}) \vee \mathrm{r}]$ $\quad \equiv(\mathrm{p} \wedge \mathrm{q}) \vee \sim \mathrm{r}$ $[(5 < 7)$ and $(7>2)]$ and $(5 \leq 2)$

Asked in: MHT CET 2020 (19 Oct Shift 1)

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