Equivalence of two implications

Statement

Determine whether $p\Rightarrow(q\Rightarrow r)$ is equivalent to $(p\land q)\Rightarrow r$.

Solution

We use the equivalence $x\Rightarrow y\equiv\lnot x\lor y$ and De Morgan’s laws. $p\Rightarrow(q\Rightarrow r)\equiv \lnot p\lor(\lnot q\lor r)\equiv \lnot p\lor\lnot q\lor r.$ $(p\land q)\Rightarrow r\equiv \lnot(p\land q)\lor r\equiv (\lnot p\lor\lnot q)\lor r\equiv \lnot p\lor\lnot q\lor r.$ The two formulae reduce to the same expression: they are equivalent. $\boxed{\text{Sì}}$

Links

Topics: Set theory
Concepts: Logical equivalence · Implication
Skills: Proving
Exercise type: Proof