Question

Need help with #6 don’t know how to find truth values.

Answer 1

The statement p is: Saturn is a planet.

The statement q is: Hong Kong is a city.

Now the first part of our composite proposition is:

`p\lor\sim q`

This means that we have to negate the statement q, this would be: Hong Kong is not a city.

Now we make the disjunction between the statement p and the negation of q, then we have:

`p\lor\sim q\text{ means: Saturn is a planet or Hong Kong is not a city}`

The second part of our composite porposition is:

`\sim p\wedge q`

This means that we have to negate the statement p, then we have: Saturn is not a planet. Then we make the conjunction between the negation of p and q, then we have:

`\sim p\wedge q\text{ means: Saturn is not a planet and Hong Kong is a city}`

Finally we make the disjunction between the statements discussed so far, hence the statament

`(p\lor\sim q)\lor(\sim p\wedge q)`

means:

Saturs is a planet or Hon kong is not a city OR Saturn is not a planet and Hong Kong is not a city.

Now, to determine the truth value of the composite proposition we have to remember that a disjunction is TRUE if one of the statements that make it is true and that the conjunction is TRUE only if both stataments that make it are true.

The first statement is: Saturn is a planet or Hong kong is not a city. Since this is a disjunction and the statement Saturn is a planet is TRUE, then the proposition is true.

The second statement is: Saturn is not a planet and Hong Kong is not a city. Since this is a conjunction and the statement Saturs is not a planet is FALSE, the the second statement is FALSE.

Now since the composite statemtent is made of a TRUE and a FALSE statement and it is a disjunction we conclude that the truth value of the statement given is TRUE.