Using laws of logic solve the following compound propositions. Also indicate the names of laws. [𝑝 ∧ (¬𝑝 ∨ 𝑞)] ∨ [𝑞 ∧ ¬(𝑝 ∧ 𝑞)]

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Let us use laws of logic solve the following compound propositions.

$[𝑝 ∧ (¬𝑝 ∨ 𝑞)] ∨ [𝑞 ∧ ¬(𝑝 ∧ 𝑞)]$

| use the distributive law |

$=[(𝑝 ∧ ¬𝑝) ∨(p\land 𝑞)] ∨ [𝑞 ∧ ¬(𝑝 ∧ 𝑞)]$

| use de Morgan law |

$=[(𝑝 ∧ ¬𝑝) ∨(p\land 𝑞)] ∨ [𝑞 ∧ (¬𝑝 \lor\neg 𝑞)]$

| use the law of contradiction |

$=[F ∨(p\land 𝑞)] ∨ [𝑞 ∧ (¬𝑝 \lor\neg 𝑞)]$

| use the distributive law |

$=[F ∨(p\land 𝑞)] ∨ [(𝑞 ∧ ¬𝑝) \lor(q\land \neg 𝑞)]$

| use the constant law |

$=(p\land 𝑞) ∨ [(𝑞 ∧ ¬𝑝) \lor(q\land \neg 𝑞)]$

| use the law of contradiction |

$=(p\land 𝑞) ∨ [(𝑞 ∧ ¬𝑝) \lor F]$

| use the constant law |

$=(p\land 𝑞) ∨ (𝑞 ∧ ¬𝑝)$

| use the commutative law |

$=(q\land p) ∨ (𝑞 ∧ ¬𝑝)$

| use the distributive law |

$=q\land (p ∨ ¬𝑝)$

| use the law of the excluded third |

$=q\land T$

| use the constant law |

$=q$