Calculus:.......6. (5 pts) Determine whether the argument is valid or invalid.?

Question

6. (5 pts) Determine whether the argument is valid or invalid.

A tree has green leaves or the tree does not produce oxygen.

This tree has green leaves

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∴ This tree does not produce oxygen.

7. (4 pts) Use Euler Diagrams to determine whether the following syllogism is valid or invalid.

Some senators are conservationists

No conservationists are pro-development

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∴ Some senators are not pro-development

8. (4 pts) Determine the truth value of the statement ~q ↔ [ ~ r ∧ (p ∨ q)] when p is false, q is true, and r is true.

9. (3pts) Determine the truth value of the following statement:

Rembrandt was a famous painter and some odd number are prime.

10. (3 pts) Use De Morgan’s Laws to determine whether the two statements are equivalent

~ (p ∧ q), ~ (q ∨ ~p)

Answer 1

6)

1. A tree has green leaves or the tree does not produce oxygen.
2. This tree has green leaves

∴ This tree does not produce oxygen.

Invalid argument.
Counterexample: A tree has green leaves and produces oxygen. As you can see, this counterexample means (1) is true, (2) is true, and (3) is false. A set of true premises and a false conclusion is an invalid argument.

8. ~q ↔ [ ~ r ∧ (p ∨ q)]
p = False
q = True
r = true

~F ↔ [ ~T ^ (F v T)].

Let's reduce them one by one. First, False or True = True.

~F ↔ [~T ^ (T)]

Let's negate each truth value now, with the ~.

T ↔ [F ^ T]

For conjunctions (^, or AND, F ^ T = F).

T ↔ F

And, by the properties of biconditionals, this is false.

F

10. Determine if ~(p ^ q) is equivalent to ~(q v ~p).

~(p ^ q) = ~p ^ ~q

~(q v ~p) = ~q v ~~p = ~q v p

They are not equivalent.

Answer 2

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