Question

Let \(U\) be the set of all problems on a comprehensive list of problems in science. Define four predicates over \(U\) by: \(P(x): x\) is a mathematics problem \(Q(x): x\) is difficult (according to some well-defined criterion: it does not matter for us What the criterion is) \(R(x): x\) is easy (according to some well-defined criterion) \(S(x): x\) is unsolvable (if you do not know what "unsolvable" means, do not worry about it here) Translate into English sentences each of the following formulas: (a) \(\forall x P(x)\) (b) \(\exists x Q(x)\) (c) \(\forall x(Q(x) \vee R(x))\) (d) \(\forall x(S(x) \rightarrow P(x))\) (e) \(\exists x(S(x) \wedge \neg P(x))\) (f) \(\neg(\forall x(\neg R(x) \vee S(x)))\) (g) \(\forall x(P(x) \rightarrow(Q(x) \leftrightarrow \neg R(x)))\) (h) \(\forall x \neg S(x)\) (i) \(\forall x(P(x) \rightarrow \neg S(x))\) (j) \(\forall x(P(x) \rightarrow(R(x) \vee S(x)))\) (k) \(\exists x(\neg Q(x) \wedge \neg R(x))\) (l) \(\exists x(R(x) \wedge S(x))\) (m) \( \forall x(Q(x) \leftrightarrow \neg R(x))\)

Short answer

Translate each formula into English sentences based on their logical meanings.

Step-by-step solution

Translate Formula A

For the formula \( \forall x P(x) \), it is stating that for every problem \( x \) in the set \( U \), \( x \) is a mathematics problem. The translation is: "All problems are mathematics problems."

Translate Formula B

The formula \( \exists x Q(x) \) indicates that there exists some problem \( x \) in the set \( U \) that is difficult. The translation is: "There is at least one problem that is difficult."

Translate Formula C

For \( \forall x(Q(x) \vee R(x)) \), it means that for every problem \( x \), \( x \) is either difficult or easy. The translation is: "Every problem is difficult or easy."

Translate Formula D

The formula \( \forall x(S(x) \rightarrow P(x)) \) suggests that if a problem \( x \) is unsolvable, then it is a mathematics problem. The translation is: "If a problem is unsolvable, then it is a mathematics problem."

Translate Formula E

The expression \( \exists x(S(x) \wedge eg P(x)) \) means there exists some problem \( x \) that is unsolvable and not a mathematics problem. The translation is: "There is at least one problem that is unsolvable and not a mathematics problem."

Translate Formula F

For \( eg(\forall x(eg R(x) \vee S(x))) \), it negates the statement that every problem is either not easy or unsolvable. The translation is: "There is at least one problem that is easy and solvable."

Translate Formula G

The formula \( \forall x(P(x) \rightarrow(Q(x) \leftrightarrow eg R(x))) \) states that for all problems \( x \), if \( x \) is a mathematics problem, then \( x \) is difficult if and only if \( x \) is not easy. The translation is: "For every mathematics problem, it is difficult if and only if it is not easy."

Translate Formula H

For \( \forall x eg S(x) \), it states that no problem \( x \) is unsolvable. The translation is: "No problem is unsolvable."

Translate Formula I

The formula \( \forall x(P(x) \rightarrow eg S(x)) \) means every mathematics problem \( x \) is not unsolvable. The translation is: "Every mathematics problem is solvable."

Translate Formula J

The expression \( \forall x(P(x) \rightarrow(R(x) \vee S(x))) \) indicates that for every mathematics problem \( x \), \( x \) is either easy or unsolvable. The translation is: "Every mathematics problem is easy or unsolvable."

Translate Formula K

For \( \exists x(eg Q(x) \wedge eg R(x)) \), it means there exists some problem \( x \) that is neither difficult nor easy. The translation is: "There is at least one problem that is neither difficult nor easy."

Translate Formula L

The expression \( \exists x(R(x) \wedge S(x)) \) suggests that there exists some problem \( x \) that is both easy and unsolvable. The translation is: "There is at least one problem that is easy and unsolvable."

Translate Formula M

For \( \forall x(Q(x) \leftrightarrow eg R(x)) \), every problem \( x \), is difficult if and only if it is not easy. The translation is: "A problem is difficult if and only if it is not easy."

Key concepts

Quantifiers

In predicate logic, quantifiers are crucial in specifying the quantity of specimens in the domain of discourse that satisfy a given property. The two primary types of quantifiers are the universal quantifier and the existential quantifier. The universal quantifier is denoted by \( \forall \) and is used to indicate that a property is true for all elements in a given set. For instance, the statement \( \forall x P(x) \) translates to "All problems are mathematics problems." This suggests that every element in the domain satisfies the condition \( P(x) \).
Conversely, the existential quantifier, represented by \( \exists \), asserts the existence of at least one element in a set that fulfills a specific property. The expression \( \exists x Q(x) \) means "There is at least one problem that is difficult," signaling that there is at least one element for which the predicate \( Q(x) \) holds true.
Understanding these quantifiers is essential for forming precise mathematical statements and is foundational to mathematical logic and reasoning.

Propositional Logic

Propositional logic is the branch of logic that deals with propositions and their logical combinations and relations. A proposition is a statement that can either be true or false. For example, "The sky is blue" is a proposition that might be true in some contexts.
In the context of our exercise, propositions can be combined using logical connectives, such as "and" (\( \wedge \)), "or" (\( \vee \)), "not" (\( eg \)), "if...then" (\( \rightarrow \)), and "if and only if" (\( \leftrightarrow \)).

• The conjunction \( Q(x) \wedge R(x) \) signifies that both predicates are true for \( x \).
• The disjunction \( Q(x) \vee R(x) \) indicates that at least one of the predicates holds.
• The negation \( eg R(x) \) implies that \( R(x) \) is false.
• Implication \( S(x) \rightarrow P(x) \) expresses that if \( S(x) \) is true, then \( P(x) \) must also be true.

The clarity in forming propositions and understanding these connectives is vital for logical operations and reasoning.

Mathematical Reasoning

Mathematical reasoning involves the process of drawing logical conclusions from premises we know to be true or assume to be true. This is a foundational skill in mathematics, allowing for the construction and verification of mathematical arguments.
Consider the exercise statement \( \forall x(S(x) \rightarrow P(x)) \): "If a problem is unsolvable, then it is a mathematics problem." This statement uses logical implication, demonstrating reasoning about the nature of problems in a logical framework.
Steps in mathematical reasoning include identifying facts, observing patterns, making conjectures, and using deductive logic. Deductive reasoning starts with a general statement and arrives at a specific conclusion, as seen when we deduce that "Every mathematics problem is solvable" from \( \forall x(P(x) \rightarrow eg S(x)) \).
The ability to apply logical operations and reason through complex problems is a hallmark of strong mathematical proficiency.

Logical Equivalence

Logical equivalence occurs when two expressions or statements are true in every possible evaluation scenario. In other words, they always have the same truth value. Logical equivalence is akin to saying two sentences mean the same thing, though they may be expressed differently.
An example from our exercise is \( \forall x(Q(x) \leftrightarrow eg R(x)) \), which declares that a problem is difficult if and only if it is not easy. This is a statement of equivalence between difficulty and the negation of easiness for all problems.
Logical equivalences can be proven by truth tables, logical identities, or algebraic manipulation of logical expressions. Recognizing these equivalences is crucial because they allow simplification of complex logical statements, making deductions clearer.
Through mastering logical equivalence, you can transform expressions into simpler or more useful forms without changing their meaning, facilitating better problem solving in mathematics.