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.
- \(A(\text{AUMAT 250})\)
- \((\exists x)A(x)\)
- \((\forall x)A(x)\)
- \(\neg (\forall x)A(x)\)
- \((\exists x)\neg A(x)\)
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.
- The instructor for this course is an excellent professor.
- Every professor at your university is excellent.
- Some professor at your university is excellent.
- Some professors at your university are excellent.
- There is at least one professor at your university who is excellent.
- Some professor at your university is not excellent.
- Some professors at your university are not excellent.
- Any professor at your university is excellent.
- 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.
- \(P(25,-1)\)
- \(P(30,-1)\)
- \(P(100,2)\lor P(100,3)\)
- \(P(100,2) \land P(100,3)\)
- \((\exists m)(\exists n) P(m,n)\)
- \((\forall m)(\forall n) P(m,n)\)
- \((\forall m)(\exists n) P(m,n)\)
- \((\exists m)(\forall n) P(m,n)\)
- \((\forall m)(\exists q)(\forall n)(P(q,n)\rightarrow P(m,n))\)
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.
- \((\exists f)(\exists g) P(f,g)\)
- \((\forall f)(\forall g) P(f,g)\)
- \((\forall f)(\exists g) P(f,g)\)
- \((\exists f)(\forall g) P(f,g)\)
- \((\forall g)(\exists f) P(f,g)\)
- \((\exists g)(\forall f) P(f,g)\)
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Consider the statement “every odd number is either \(1\) more or \(3\) more than a mulitple of \(4\text{.}\)”
- Assign appropriate predicates (with domains explictly stated), and then translate the statement into symbolic logic.
- Negate the statement, and simplify the logical expression so that any/all negation symbols appear directly to the left of a predicate.
- Translate your simplified negated statement from Task b into English.
- 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
- Translate the statement into English.
- Determine whether the statement is true.
- 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 \(E\text{.}\)
- Translate your simplified negated statement from Part c into English.
- 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 \(x\text{.}\)
- All university students study diligently.
- 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”.
- \((\forall x)[ U(x) \rightarrow S(x) ]\text{.}\)
- \((\exists x)[ U(x) \rightarrow S(x) ]\text{.}\)
Are the student's answers correct? Justify your assessment.
- Hint
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Try translating the student's symbolic language statements back into English, explicitly using the stated domain of \(x\) , and see what you get.