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Translate the following English sentences into the symbolic language. Use S for 'Smith is here' and J for (Jones is away': 1. Jones is not away if and only if Smith is not away 2. Jones is away if and only if Smith is here, but Jones is not away 3. It is not the case that both Smith is here and Jones is not away 4. Either Jones is away or if Smith is here then Jones is not away 5. Smith is here, and Jones is away or Jones is not away 6. Jones is away, and it is not the case that either Smith is here or Jones is away Use the method of truth tables to determine whether or not each of the following arguments is valid: 7. (B=A) 8. B D A - A B A B A (B. A) .. A Use the method of formal proof together with the 6 elementary rules of inference [MP, MT, Add, D.S., Simp, Conj] to demonstrate the validity of the following arguments: 9. A u B 10. A C BvC AvB A / C Use the method of formal proof again but this time add the three replacement rules [DN, Com, Assoc] to the 6 rules above to demonstrate the validity of the following arguments: 11.B.A 12. (A.B).C. C (BVC)vD A u D/AC.D ~B

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