Problem: Symbolizing Sentences Using Quantifiers

We are tasked to:

1. Explain quantifiers.
2. Symbolize given sentences (i) with a universe of discourse and (ii) without specifying a universe of discourse.

Step 1: Understanding Quantifiers

Quantifiers are symbols used in logic to express "how many" elements of a set satisfy a condition. The two main quantifiers are:

1. Universal Quantifier ($\forall$): "For all" or "Every".
   • Example: $\forall x P(x)$: For all $x$, $P(x)$ holds.
2. Existential Quantifier ($\exists$): "There exists" or "Some".
   • Example: $\exists x P(x)$: There exists at least one $x$ such that $P(x)$ holds.

Step 2: Symbolizing the Sentences

Sentence (1): "All states in India are not highly populated."

• Interpretation: For all states $x$ in India, it is not true that $x$ is highly populated.
• Let $P(x)$: $x$ is highly populated.
• Universe of discourse: All states in India.

1. With Universe of Discourse:
   $\forall x \neg P(x).$
   Meaning: "For all $x$, $P(x)$ is not true" (no state is highly populated).

2. Without Universe of Discourse:
   To explicitly state the universe of discourse (states in India), we write:
   $\forall x (S(x) \implies \neg P(x)),$
   where $S(x)$: $x$ is a state in India.

   • Meaning: "For all $x$, if $x$ is a state in India, then $x$ is not highly populated."

Sentence (2): "Some students are tall but not healthy."

• Interpretation: There exist students $x$ such that $x$ is tall and $x$ is not healthy.
• Let $T(x)$: $x$ is tall, and $H(x)$: $x$ is healthy.
• Universe of discourse: All students.

1. With Universe of Discourse:
   $\exists x (T(x) \land \neg H(x)).$
   Meaning: "There exists an $x$ such that $x$ is tall and $x$ is not healthy."

2. Without Universe of Discourse:
   To explicitly state the universe of discourse (students), we write:
   $\exists x (S(x) \land T(x) \land \neg H(x)),$
   where $S(x)$: $x$ is a student.

   • Meaning: "There exists an $x$ such that $x$ is a student, $x$ is tall, and $x$ is not healthy."

Final Answer

1. Sentence 1:

   • With universe of discourse: $\forall x \neg P(x)$.
   • Without universe of discourse: $\forall x (S(x) \implies \neg P(x))$.

2. Sentence 2:

   • With universe of discourse: $\exists x (T(x) \land \neg H(x))$.
   • Without universe of discourse: $\exists x (S(x) \land T(x) \land \neg H(x))$.

Explanation Summary

• Quantifiers allow us to express "all" ($\forall$) or "some" ($\exists$) in logical form.
• Universe of discourse defines the set over which variables ($x$) range.
• If the universe is not explicitly defined, we include an additional condition (e.g., $S(x)$: $x$ is a state or student).

Related Questions

1. What is the difference between $\forall$ and $\exists$?
2. How do you symbolize statements with "not" using quantifiers?
3. What is the role of the universe of discourse in logic?
4. How do you negate quantified statements?
5. Can you give an example of using both $\forall$ and $\exists$ in a single statement?

Tip: Always define your predicates clearly, such as $P(x)$ or $T(x)$, to avoid confusion when symbolizing statements.