4.5: Exercises

Interpreting symbolic language

Let \(A(x)\) represent the predicate “\(x\) is a wonderful learning experience”, where \(x\) is a free variable in the domain of all university courses.

Translate each of the following into an English sentence that is grammatically correct.

1. \(A(\text{AUMAT 250})\)
2. \((\exists x)A(x)\)
3. \((\forall x)A(x)\)
4. \(\neg (\forall x)A(x)\)
5. \((\exists x)\neg A(x)\)

Translating into symbolic language.

Let \(B(x)\) represent the predicate “\(x\) is excellent”, where \(x\) is a free variable in the domain of all Augustana professors.

Translate each of the following into symbolic language.

6. The instructor for this course is an excellent professor.
7. Every professor at your university is excellent.
8. Some professor at your university is excellent.
9. Some professors at your university are excellent.
10. There is at least one professor at your university who is excellent.
11. Some professor at your university is not excellent.
12. Some professors at your university are not excellent.
13. Any professor at your university is excellent.
14. No professor at your university is excellent.

Analyzing predicate statements about integers

Let \(P(m,n)\) represent the predicate \(2m - 45n > 101\text{,}\) where \(m\) and \(n\) are free variables in the domain of integers.

For each of the following, determine whether the statement is true or false. Explain your reasoning.

15. \(P(25,-1)\)
16. \(P(30,-1)\)
17. \(P(100,2)\lor P(100,3)\)
18. \(P(100,2) \land P(100,3)\)
19. \((\exists m)(\exists n) P(m,n)\)
20. \((\forall m)(\forall n) P(m,n)\)
21. \((\forall m)(\exists n) P(m,n)\)
22. \((\exists m)(\forall n) P(m,n)\)
23. \((\forall m)(\exists q)(\forall n)(P(q,n)\rightarrow P(m,n))\)

Analyzing predicate statements about functions

(Requires calculus.) Let \(P(f,g)\) represent the predicate \(\frac{df}{dx} = g\text{,}\) where \(f\) and \(g\) are free variables in the domain of continuous functions in the real variable \(x\text{.}\)

For each of the following, determine whether the statement is true or false. Explain your reasoning.

24. \((\exists f)(\exists g) P(f,g)\)
25. \((\forall f)(\forall g) P(f,g)\)
26. \((\forall f)(\exists g) P(f,g)\)
27. \((\exists f)(\forall g) P(f,g)\)
28. \((\forall g)(\exists f) P(f,g)\)
29. \((\exists g)(\forall f) P(f,g)\)

30. Consider the statement “every odd number is either \(1\) more or \(3\) more than a mulitple of \(4\text{.}\)”

    1. Assign appropriate predicates (with domains explictly stated), and then translate the statement into symbolic logic.
    2. Negate the statement, and simplify the logical expression so that any/all negation symbols appear directly to the left of a predicate.
    3. Translate your simplified negated statement from Task b into English.

31. Let \(P(f,g)\) represent the predicate \(\dfrac{df}{dx} = g\text{,}\) and let \(E(f,g)\) represent the predicate \(g = f\text{,}\) where \(f\) and \(g\) are free variables in the domain of functions in the real variable \(x\text{.}\) Consider the statement

\begin{equation*} (\forall f)(\forall g)\{(\exists h)\{P(f,h) \land P(g,h)\} \rightarrow E(f,g)\} \text{.} \end{equation*}

1. Translate the statement into English.
2. Determine whether the statement is true.
3. Working with the originally provided symbolic version above, negate the statement. Simplify the negated version to so that any/all negation symbols appear directly to the left of one of the predicates \(P\) or \(E\text{.}\)
4. Translate your simplified negated statement from Part c into English.

32. You've become an expert at predicate logic, and now make a (very meagre) living grading logic assignments for a large university. Here is the question you've been assigned to mark two thousand times.

Let \(x\) represent a free variable from the domain of all living humans.

Translate the following two statements into properly quantified predicate statements in the variable \(x\text{.}\)

1. All university students study diligently.
2. Some university students study diligently.

You pick up the first assignment. Here is the student's answer.

Let \(U(x)\) mean “\(x\) is a university student”. Let \(S(x)\) mean “\(x\) studies diligently”.

1. \((\forall x)[ U(x) \rightarrow S(x) ]\text{.}\)
2. \((\exists x)[ U(x) \rightarrow S(x) ]\text{.}\)

Are the student's answers correct? Justify your assessment.

Hint

Try translating the student's symbolic language statements back into English, explicitly using the stated domain of \(x\), and see what you get.