# Homework Help: Quantified Statements

1. Jul 2, 2013

### Bashyboy

1. The problem statement, all variables and given/known data
A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, and 1 computer science major who is a senior. Express each of these statements in terms of quantifiers and then determine its truth value.

a) There is a student in the class who is a junior.

b)Every student in the class is a computer science major.

c) There is a student in the class who is neither a mathematics major nor a junior.

d)Every student in the class is either a sophomore or a computer science major.

e) There is a major such that there is a student in the class in every year of study with that major.

2. Relevant equations

3. The attempt at a solution

First of all, let P(s, c, m) be "student s has class standing c and is majoring in m."

I am having trouble with part (c). My answer to this part is $\exists s \exists c \exists m (\neg P(s,c,math) \wedge \neg P(s,junior,m)$ Evidently, the true anser is $\exists s \exists c \exists m (P(s,c,m) \wedge (c ~ \ne ~ junior) \wedge (m ~ \ne ~ math)$ At first I figured what was wrong with my answer was, that P(s,c,math) and P(s,junior,m) spoke about two different students; but then I realized, since both propositional functions assumed the variable s, they must be speaking of a single person at a time. Translating my answer to English, "It is not true that student s has class standing c and is majoring in math, and it is not true that this same student is a junior and is majoring in m," which I imagine would simplify to, and be logically equivalent to, "There is a student in the class who is neither a mathematics major nor a junior."

So, is my answer equally valid?

Last edited: Jul 2, 2013
2. Jul 2, 2013

### verty

Nope, your answer is weaker. You can split your answer into two because c appears only in the first half and m only in the second half.

Split it into two, translate that into English and you should see why it is weaker.

3. Jul 2, 2013

### D H

Staff Emeritus
Suppose all of the comp sci majors taking the class are juniors as opposed to a mix of sophomores, juniors, and seniors. With this change, the statement "there is a student in the class who is neither a mathematics major nor a junior" is false. The correct answer also is false in this circumstance.

Now let's look at your representation of the statement. Set s to be one of those comp sci majors, c to be any year, and m to be math. Then P(s,c,math) both P(s,junior,m) are both false. Thus $\exists s \exists c \exists m (\neg(P(s,c,\text{math}) \wedge \neg P(s,\text{junior},m))$ is true.

4. Jul 2, 2013

### Bashyboy

So, DH, do you consent to my answer being correct? If so, I do have the notion that the answer the book provides is much simpler in terms of comprehensibility.

5. Jul 2, 2013

### D H

Staff Emeritus