Solution to Problem 2.24 ECS 101 Lab 4

Solution to 2.24 Lets look at a problem that includes: Connectives in Compound Statements Arguments using previous Conclusions

Do Martians have three heads? Are they green? Can they fly? When the first astronaut to visit the planet Mars returned to Earth, he was asked to describe the inhabitants of the “red planet." Still suffering from the effects of interplanetary travel, he answered in the following correct, but confusing, manner:  "It is not true that if Martians are green then they either have three heads or else they cannot fly, unless it is also true that they are green if and only if they can fly and that they do not have three heads."  Assume that all Martians look alike and that they have at least one of the three characteristics referred to.

How do we solve the problem? Lets define a set of statements whose truth values will represent the solution. g : Martians are green. h : Martians have three heads. f : Martians can fly. Now take the astronaut’s statement and represent it symbolically:

Use a Symbolic Representation "It is not true that if Martians are green (g) then (->) they either have three heads (h) or else they cannot fly (~f), unless it is also true that (->) they are green (g) if and only if (<->) they can fly (f) and that they do not have three heads (~h).“ [ g -> ( h OR ( ~f ) ) ] -> [ ( g <-> f ) AND ( ~h ) ] THIS IS A COMPOUND STATEMENT.

Use Previous Conclusions Looking at the expression: [ g -> ( h OR ( ~f ) ) ] -> [ ( g <-> f ) AND ( ~h ) ] From Conclusion If 4: Either [ g -> ( h OR ( ~f ) ) ] is false, Or [ ( g <-> f ) AND ( ~h ) ] is true.

If [ g -> ( h OR ( ~f ) ) ] is false From Conclusion If 5: g must be true, and h OR (~f) must be false. For the second result we use Condition Or 1: h must be false. (~f) must be false. Therefore, f must be true.

If [ ( g <-> f ) AND ( ~h ) ] is true From Conclusion And 1: g <-> f must be true, and (~h) must be true. Therefore h must be false. For the first result, we use Condition Bi 1: g must be true. f must be true.

Compare Results Both alternatives have the same results: g must be true. h must be false. f must be true. So we have solved the problem: Martians are green. Martians do not have three heads. Martians can fly.