Question

Let the universe $U$ be the set of all human beings living in the year $2001,$ and translate the following English sentences into quantified formulas. Let $P(x)$ stand for $"x$ is young" $Q(x)$ for ${}^{4}x$ is female," and $R(x)$ for ${}^{\ast }x$ is an athlete." (a) "All athletes are young." (b) "Not all young people are athletes." (c) "All young people are not athletes." (Warning: In informal English, this sentence has two quite different meanings. One is "more grammatically correct" than the other, however, and that is the one we're asking for.) (d) "Some young people are not athletes." (e) "Some athletes are young females." (f) "All athletes are young males." (g) "Some athletes are female and are not young." (h) "Some young females are not athletes." (i) "All young females are athletes." (j) "Some athletes are not young." (k) "No young people are athletes." (1) "All athletes are either female or are young." (m) "If all athletes are female, then all athletes are young; otherwise, no athletes are young."

Short answer

Quantified expressions range from $\mathrm{\forall }x(R(x)\to P(x))$ to complex conditionals like in Step 13.

Step-by-step solution

Translate 'All athletes are young'

To represent the sentence "All athletes are young," we need to express that for every human being, if they are an athlete, then they are young. In logical notation, this becomes:$$\mathrm{\forall }x(R(x)\to P(x))$$

Translate 'Not all young people are athletes'

This sentence means there exists at least one young person who is not an athlete. Use the existential quantifier to express this:$$\mathrm{\exists }x(P(x)\wedge egR(x))$$

Translate 'All young people are not athletes'

This grammatically correct interpretation is equivalent to saying that no young person is an athlete: $$\mathrm{\forall }x(P(x)\to egR(x))$$

Translate 'Some young people are not athletes'

This sentence implies that there exists a young person who is not an athlete:$$\mathrm{\exists }x(P(x)\wedge egR(x))$$

Translate 'Some athletes are young females'

This statement means there exists someone who is both an athlete and a young female. Express this as:$$\mathrm{\exists }x(R(x)\wedge P(x)\wedge Q(x))$$

Translate 'All athletes are young males'

This means if someone is an athlete, they must be both young and male. In logical form:$$\mathrm{\forall }x(R(x)\to (P(x)\wedge egQ(x)))$$

Translate 'Some athletes are female and are not young'

This sentence suggests there is at least one athlete who is female and not young:$$\mathrm{\exists }x(R(x)\wedge Q(x)\wedge egP(x))$$

Translate 'Some young females are not athletes'

This indicates the existence of someone who is both young and female, and not an athlete:$$\mathrm{\exists }x(P(x)\wedge Q(x)\wedge egR(x))$$

Translate 'All young females are athletes'

This means every young female is an athlete, which is represented as:$$\mathrm{\forall }x((P(x)\wedge Q(x))\to R(x))$$

Translate 'Some athletes are not young'

There exists at least one athlete who is not young:$$\mathrm{\exists }x(R(x)\wedge egP(x))$$

Translate 'No young people are athletes'

This implies that for any individual, if they are young, they cannot be an athlete:$$\mathrm{\forall }x(P(x)\to egR(x))$$

Translate 'All athletes are either female or are young'

Every athlete is either female or young, written as:$$\mathrm{\forall }x(R(x)\to (Q(x)\vee P(x)))$$

Translate 'If all athletes are female, then all athletes are young; otherwise, no athletes are young'

This is a conditional statement involving a consequence if one premise holds, and otherwise a different outcome. It translates as:$$((\mathrm{\forall }x(R(x)\to Q(x)))\to \mathrm{\forall }x(R(x)\to P(x)))\wedge (eg(\mathrm{\forall }x(R(x)\to Q(x)))\to \mathrm{\forall }x(R(x)\to egP(x)))$$

Key concepts

Logical Notation

Logical notation is a symbolic way to represent logical expressions clearly and concisely. These symbols are the language of logic, much like mathematical symbols are used in equations.

Understanding logical notation can help us translate everyday language into formal expressions, making logical arguments easier to analyze and understand. Here are some common symbols used in logical notation:

• $\mathrm{\forall }$ - Universal quantifier, meaning "for all" or "for every."
• $\mathrm{\exists }$ - Existential quantifier, meaning "there exists."
• $\wedge$ - Logical conjunction, meaning "and."
• $\vee$ - Logical disjunction, meaning "or."
• $\to$ - Logical implication, meaning "implies."
• $eg$ - Logical negation, meaning "not."

Using these symbols, we can take complex logical statements from English and distill them into precise mathematical forms. This help us in understanding and proving logical theorems or arguments.

Universal Quantification

Universal quantification is essential when you want to express that something is true for all members of a specific set or group. It uses the symbol $\mathrm{\forall }$, which stands for "for all" or "for every."

The key idea here is to make a statement that applies universally within the scope of concern. For instance, the statement "All athletes are young" can be formulated in logical notation as $\mathrm{\forall }x(R(x)\to P(x))$. This expression means that for every element $x$ in the universe, if $x$ is an athlete $R(x)$, then $x$ must also be young $P(x)$.

Universal quantification is powerful because it allows us to make sweeping generalizations about a set of objects, granting them certain properties. Understanding this concept is foundational for valid logical reasoning and proof deduction in mathematics and various fields.

Existential Quantification

Existential quantification focuses on expressing that there is at least one element in a particular set for which a given statement is true. It uses the symbol $\mathrm{\exists }$, which stands for "there exists."

This concept is often employed when we want to assert that something specifically exists within the universe of discourse. For instance, "Some young people are athletes" is an example involving existential quantification. This would be formulated as $\mathrm{\exists }x(P(x)\wedge R(x))$. It indicates that there is at least one individual $x$ who is both young $P(x)$ and an athlete $R(x)$.

Existential quantification allows us to pinpoint the existence of certain characteristics or conditions within a broader range, helping define and argue particular hypotheses or real-world situations by focusing on individual instances rather than a whole group.