Question

1. For each of the following, state whether it is true or false. If false, give a counterexample. If true, give a convincing argument why it is true. For all a, B e Pimla, (a) if (a + B) is valid and a is valid, then ß is valid. (b) if (a + b) is satisfiable and a is satisfiable, then ß is satisfiable. (c) if (a + b) is valid and a is satisfiable, then ß is satisfiable.

Answer 1

Answerian Given that, nal for each of the following state whether it is true or false. If false, give a Counter example, If true give a convincing argument why it is true. For all d,BE Pero a) if (d>B) is valid and dis valid, . then ß is valid => Here the wood "Valid" means in all - the cases it should be true and valid false 1:2 is always true => ß Here d = true ß= toue it is I always true. > >ß Here d=true ß-alse it is from the above given statement it shows that a is always tome so B should be true. i dß must be true and valid. i dß is valid sis valid and that ß is valid

b) if (2 B ) is satisfiable and a is Satisfiable , then ß is satisfiable. => Here satisfiable means it should satisfy minimum One True value as input d ß 2ß Tone True True (satisfiable) false false Tone / satisfiable) True False false false Touer True (satisfiable) -> Here from above data when dis True and B is false 2Jß is false it means unsatisfiable But in other three cases it is satisfiable. So, we need Only minimum Tone value at least one also sufficient to be Satisfy. i 2 B is satisfiable , a is satisfiable then B is satisfiable. Be Guse B also have Tome values.

() if (@>B) is valid and dis satisfiable. - then ß is satisfiable. Here "valid" means all the cases it should be true and satisfiable" means atleast one Case can be true. :So we have to show that d->ß is valid. d B LAB Toner Tone (valid) false false Tone (valid) False Toner Tone. (valid) : In the above all Cases dəß is True and it valid. i2 has one True value it is Satis fiable and ß has a Tome values i t is also satisfiable. Tone,