For propositions p, q, and r, determine whether 𝑝 ⟶ (𝑞 ⟶ 𝑟) and (𝑝 ⟶ 𝑞) ⟶ 𝑟 are logically equivalent.
If ∣p∣=∣q∣=∣s∣=F,|p|=|q |=|s|=F,∣p∣=∣q∣=∣s∣=F, then ∣𝑝→(𝑞→𝑟)∣=F→(F→F)=F→T=T|𝑝 \to (𝑞 \to 𝑟)|=F \to (F \to F)=F\to T=T∣p→(q→r)∣=F→(F→F)=F→T=T but ∣(𝑝→𝑞)→𝑟∣=(F→F)→F=T→F=F|(𝑝 \to 𝑞) \to 𝑟|=(F \to F) \to F=T\to F=F∣(p→q)→r∣=(F→F)→F=T→F=F, and we conclude that the formulas are not logically equivalent.
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