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Question Number 46475 by peter frank last updated on 27/Oct/18

Commented by Joel578 last updated on 27/Oct/18

$$plsrecheckquestion\left(b\right)$$

Commented by peter frank last updated on 28/Oct/18

$$\mathrm{b})[(\mathrm{p}\to \mathrm{q})\wedge (\mathrm{q}\iff \mathrm{p})\wedge \sim \mathrm{p}]\to \sim \mathrm{q}$$

Answered by Joel578 last updated on 27/Oct/18

$$(p\to q)\wedge (q\to \sim r)\wedge r$$$$\equiv (\sim p\vee q)\wedge [(\sim q\vee \sim r)\wedge r]$$$$\equiv (\sim p\vee q)\wedge [(r\wedge \sim q)\vee (r\wedge \sim r)]$$$$\equiv (\sim p\vee q)\wedge [(r\wedge \sim q)\vee F]$$$$\equiv (\sim p\vee q)\wedge (r\wedge \sim q)$$$$\equiv [(\sim p\vee q)\wedge \sim q]\wedge r$$$$\equiv [(\sim q\wedge \sim p)\vee (\sim q\wedge q)]\wedge r$$$$\equiv [(\sim q\wedge \sim p)\vee F]\wedge r$$$$\equiv (\sim q\wedge \sim p)\wedge r$$$$$$$$[(\sim q\wedge \sim p)\wedge r]\to \sim p$$$$\equiv \sim [(\sim q\wedge \sim p)\wedge r]\vee \sim p$$$$\equiv [(q\vee p)\vee \sim r]\vee \sim p$$$$\equiv q\vee (p\vee \sim p)\vee \sim r$$$$\equiv q\vee T\vee \sim r$$$$\equiv T$$

Commented by peter frank last updated on 28/Oct/18

$$\mathrm{thanks}$$

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