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If p, $q$ are true statement and $\mathrm{r}$ is false statement, then which of the following atements is a true statement.
Options:
Solution:
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Verified Answer
The correct answer is:
$(p \leftrightarrow q) \leftrightarrow r$ is false.
$(\mathrm{A})$
We have $\mathrm{p} \equiv \mathrm{T}, \mathrm{q} \equiv \mathrm{T}$ and $\mathrm{r} \equiv \mathrm{F}$
We will check truth value of each option.
(A) $(\mathrm{p} \leftrightarrow \mathrm{q}) \leftrightarrow \mathrm{r}$
$\equiv(T \leftrightarrow T) \leftrightarrow F \equiv T \leftrightarrow F \equiv F$
(B) $(p \vee q) \vee r$
$\equiv(T \vee T) \vee F \equiv T \vee F=T$
(C) $(p \wedge q) \rightarrow r$
$\equiv(\mathrm{T} \wedge \mathrm{T}) \rightarrow \mathrm{F} \equiv \mathrm{T} \rightarrow \mathrm{F} \equiv \mathrm{F}$
(D) $(\mathrm{p} \rightarrow \mathrm{r}) \rightarrow \mathrm{q}$
$\equiv(T \rightarrow F) \rightarrow T \equiv F \rightarrow T \equiv T$
We have $\mathrm{p} \equiv \mathrm{T}, \mathrm{q} \equiv \mathrm{T}$ and $\mathrm{r} \equiv \mathrm{F}$
We will check truth value of each option.
(A) $(\mathrm{p} \leftrightarrow \mathrm{q}) \leftrightarrow \mathrm{r}$
$\equiv(T \leftrightarrow T) \leftrightarrow F \equiv T \leftrightarrow F \equiv F$
(B) $(p \vee q) \vee r$
$\equiv(T \vee T) \vee F \equiv T \vee F=T$
(C) $(p \wedge q) \rightarrow r$
$\equiv(\mathrm{T} \wedge \mathrm{T}) \rightarrow \mathrm{F} \equiv \mathrm{T} \rightarrow \mathrm{F} \equiv \mathrm{F}$
(D) $(\mathrm{p} \rightarrow \mathrm{r}) \rightarrow \mathrm{q}$
$\equiv(T \rightarrow F) \rightarrow T \equiv F \rightarrow T \equiv T$
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