By giving a counterexample,show that the following statement is not true.
$p:$ If all the angles of a triangle are equal,then the triangle is an obtuse-angled triangle.
$p:$ If all the angles of a triangle are equal,then the triangle is an obtuse-angled triangle.
(N/A) The given statement is of the form 'if $q$ then $r$'.
$q:$ All the angles of a triangle are equal.
$r:$ The triangle is an obtuse-angled triangle.
The statement $p$ is false if we can find a case where $q$ is true but $r$ is false.
In an equilateral triangle,all three angles are equal to $60^{\circ}$.
Since $60^{\circ} < 90^{\circ}$,an equilateral triangle is an acute-angled triangle,not an obtuse-angled triangle.
Thus,the statement $p$ is false because we have found a counterexample (the equilateral triangle) where the premise is true but the conclusion is false.
$q:$ All the angles of a triangle are equal.
$r:$ The triangle is an obtuse-angled triangle.
The statement $p$ is false if we can find a case where $q$ is true but $r$ is false.
In an equilateral triangle,all three angles are equal to $60^{\circ}$.
Since $60^{\circ} < 90^{\circ}$,an equilateral triangle is an acute-angled triangle,not an obtuse-angled triangle.
Thus,the statement $p$ is false because we have found a counterexample (the equilateral triangle) where the premise is true but the conclusion is false.
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Similar Questions
The correct logical equivalences from the following are:
$(I)$ $p \to (q \to r) \equiv (p \land q) \to r$
$(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$
$(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$
$(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$
$(I)$ $p \to (q \to r) \equiv (p \land q) \to r$
$(II)$ $(p \to q) \to r \equiv p \to (q \lor r)$
$(III)$ $(p \to q) \to r \equiv (p \to r) \land (\sim q \to r)$
$(IV)$ $p \to (q \to r) \equiv q \to (p \to r)$
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