Answer to Question #183901 in Discrete Mathematics for Maricel

Question

RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is

used in each of the following arguments? Show solution. (4 pts each)

1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the

problem was solved. If Jen wrote a Java source code, then the problem was solved.

2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be

vaccinated. 

Answer 1

1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the problem was solved. If Jen wrote a Java source code, then the problem was solved.

Answer: Argument is valid.

p – Joy wrote a C++ source code

q – Jen wrote a Java source code

r – the problem was solved

Joy wrote a C++ source code, or Jen wrote a Java source code : p v q

If Joy wrote a C++ source code, then the problem was solved : p → r

If Jen wrote a Java source code, then the problem was solved : q → r

Here is used Rules for disjunctions, namely Case analysis:

p→r, q→r, p v q ├ r

2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be vaccinated.

Answer: Argument is valid.

p(x) – person x likes to be COVID – 19 positive

q(x) – person x likes to be vaccinated

q(x) → p(x) – if person x likes to be vaccinated, then person x likes to be COVID – 19 positive, so $\exists$x q(x) → $\exists$x p(x)

Using Transposition or contraposition law : p → q ├ ¬p → ¬q

¬($\exists$x p(x)) → ¬($\exists$x q(x))

¬($\exists$x p(x)) → $\forall$x ¬q(x)

it means given argument sentence