4.4: Activities

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Activity \(\PageIndex{1}\)

Devise an example of predicates \(A(x)\) and \(B(x)\) such that, of the statements

\((\forall x)\{A(x) \lor B(x)\}\text{,}\) and

\((\forall x)A(x) \lor (\forall x)B(x)\text{,}\)

the first is true but the second is false.

Devise an example of predicates \(A(x)\) and \(B(x)\) such that, of the statements

\((\exists x)\{A(x) \land B(x)\}\text{,}\) and
\((\exists x)A(x) \land (\exists x)B(x)\text{,}\)

the first is false but the second is true.

Activity \(\PageIndex{2}\)

Let \(P(f,g)\) represent the predicate \(\dfrac{df}{dx} = g\text{,}\) 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.

\((\exists f)(\exists g) P(f,g)\)
\((\forall f)(\forall g) P(f,g)\)
\((\forall f)(\exists g) P(f,g)\)
\((\exists f)(\forall g) P(f,g)\)
\((\forall g)(\exists f) P(f,g)\)
\((\exists g)(\forall f) P(f,g)\)

Activity \(\PageIndex{3}\)

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 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*}

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 Task c into English.

Activity \(\PageIndex{3}\)

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: Try translating the student's symbolic language statements back into English, explicitly using the stated domain of \(x\), and see what you get. Is it possible for the student's version of the statement to be true in a way that goes against the idea expressed by the original English version of the statement?