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Q.
Let $\Delta, \nabla \in\{\wedge, \vee\}$ be such that $p\, \nabla \,a \Rightarrow((p \,\nabla\, q) \nabla\, r)$ is a tautology. Then $(p \,\nabla\, q) \Delta r$ is logically equivalent to :
Case-I If $\Delta \equiv \nabla \equiv \wedge$
$( p \wedge q ) \rightarrow(( p \wedge q ) \wedge r )$
it can be false if $r$ is false,
so not a tautology
Case-II If $\Delta \equiv \nabla \equiv \vee$
$( p \vee q ) \rightarrow(( p \vee q ) \vee r ) \equiv$ tautology
then $( p \vee q ) \vee r \equiv( p \Delta r ) \vee q$
Case-III if $\Delta=\vee, \nabla=\wedge$
then $( p \wedge q ) \rightarrow\{( p \vee q ) \wedge r \}$
Not a tautology
$($ Check p $\rightarrow T , q \rightarrow T , r \rightarrow F )$
Case-IV if $\Delta=\wedge, \nabla=\vee$
$( p \wedge q ) \rightarrow\{( p \wedge q ) \vee r \}$
Not a tautology