Exercise 5 State whether the following argument is valid ...



Solutions courtesy of Jiayi Ye

Exercise 5 State whether the following argument is valid or invalid. Give reasons

Premises: All those who are drug addicts drank milk as a child. Joe drank milk as a child

Conclusion: Joe will become a drug addict

Answer:

The conclusion is false because not all the people drink milk as children are drug addicts.

[pic]

In other words, Joe might not be a drug addict, while he drank milk as a child. Therefore, the argument is invalid.

2

Exercise 6 State whether the following argument is valid or invalid. Give reasons.

Premises: If you do every problem in the book, then you will learn the subject. You learned the subject

Conclusion: You did every problem in the book.

Answer:

The argument is invalid. The reason is you can learn the subject not only by doing all the problems in the book. Here is the diagram showing the relationship between doing all the problems in the book and learn the subject.

Another way of thinking of this is as follows:

The first premise is in the form of a p ( q statement with p the statement “you do every problem in the book” and q the statement “ you learned the subject”. We know however that p ( q is not the same as the converse q ( p so that we cannot conclude that should q be true then p will also be true.

[pic]

Therefore, the argument is invalid.

2

Exercise 7 State whether the following argument is valid or invalid. Give reasons.

Premises: If it snows today, the university will close. The university is not closed today

Conclusion: It did not snow today.

Again – for p = its snows today and q = the university is closed , we have a p ( q statement, which is equivalent to the contrapositive ~q ( ~p. Therefore knowing the athe university is closed - namely ~q - it follows that ~p is also true - namely it did not snow.

The argument is valid. The premises say that if it is snows today, the university will close. This means that snow today will be one factor making the university must close; however, the university is not close today, that mean the factors of making the university close does not exist any. Here is the diagram show the relationship between these premises.

[pic]It shows that it does not snow today because the university does not close. Therefore, snows today as a factor to make the university close was not occurring, so that this argument is valid.

2

Exercise 12 Prove using truth tables that [pic] - namely a statement

p[pic]q is equivalent to its contra-positive. Why does this also show that the inverse is

equivalent to the converse?[pic]

Answer:

|p |q |[pic],([pic]) |~p |~q |[pic],([pic]) |

|T |T |T |F |F |T |

|T |F |F |F |T |F |

|F |T |T |T |F |T |

|F |F |T |T |T |T |

Because [pic]and [pic]have the same column in the truth table!

Therefore, its contraposition is equivalent.

And also, the inverse is equivalent to the converse.

From the truth tables the converse and the inverse is equivalent to each other.

|p |q |[pic],([pic]) |~p |~q |[pic],([pic]) |

|T |T |T |F |F |T |

|T |F |T |F |T |T |

|F |T |F |T |F |F |

|F |F |T |T |T |T |

Because [pic]and [pic]have the same column in the truth table; therefore, its inverse is equivalent to the converse.

2

Exercise 13 Prove using truth tables that ~ (p [pic] q) [pic] ~p [pic] ~q and ~ (p [pic] q) [pic]~

p[pic] ~ q .

Answer:

In the truth table, the (p [pic] q) is showing as follow

|p |q |[pic] |

|T |T |T |

|T |F |T |

|F |T |T |

|F |F |F |

Therefore, the ~([pic])would be as follow and the ~p [pic] ~q

|p |q |~([pic]) |~p [pic] ~q |

|T |T |F |F |

|T |F |F |F |

|F |T |F |F |

|F |F |T |T |

You can see the columns of ~p [pic] ~q and ~([pic]) are all the same, therefore, they are equivalent.

And here is the truth table of (p [pic] q), ~ (p [pic] q) and ~ p[pic] ~ q

|p |q |p [pic] q |~ (p [pic] q) |~ p[pic] ~ q |

|T |T |T |F |F |

|T |F |F |T |T |

|F |T |F |T |T |

|F |F |F |T |T |

You can see the columns of ~ (p [pic] q) and (~ p[pic] ~ q) are all the same, therefore, they are equivalent.

2

Total: 10

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download