Answer to Question #177375 in Discrete Mathematics for Zain Ul Abideen Khan

Question

For each of the arguments below, determine whether the argument is correct or incorrect and explain why.

a)     All students in this class understand logic. Xavier is a student in this class. Therefore, Xavier understands logic.

b)    Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major.

c)     Everyone who eats granola every day is healthy. Linda is not healthy. Therefore, Linda does not eat granola every day.

Answer 1

a) Lets assume:

P(x) = "x is a student in this class"

Q(x) = "x understands logic"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. P(Xavier) premise
3. $P(Xavier) \to Q(Xavier)$ universal installation from 1
4. Q(Xavier) modus ponens from 2 and 3

Step 4 means that "Xavier" understand logic" and this corresponds with the given sentence thus the argument is correct.

b) Lets assume:

P(x) = "x is a computer science major"

Q(x) = "x takes discrete mathematics"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. P(Natasha) premise
3. $P(Natasha) \to Q(Natasha)$ universal installation from 1

There is no rule for inference that allows us to conclude P(Natasha) which means "Natasha is a computer science major." and thus the argument is incorrect.

c) Lets assume:

P(x) = "x eats granola every day"

Q(x) = "x is healthy"

We can rewrite the given sentence:

1. $\forall x(P(x)\to Q(x))$ premise
2. $\neg Q(Linda)$ premise
3. $P(Linda) \to Q(Linda)$ universal installation from 1
4. $\neg P(Linda)$ modus tollens from 2 and 3

Step 4 means that "Linda does not eat granola every day." and this corresponds with the given sentence thus the argument is correct.

a) correct

b) incorrect

c) correct