If the universe of discourse 𝑈={1,2,3} then the statement ∃𝑥𝑃(𝑥) is equivalent to


If the universe of discourse \( U = \{1, 2, 3\} \), the statement \( \exists x \, P(x) \) (meaning "there exists an \( x \) such that \( P(x) \) is true") can be expanded based on the given elements of the universe.

Since \( U = \{1, 2, 3\} \), the statement \( \exists x \, P(x) \) means that \( P(x) \) holds for **at least one** element from the set \( U \). Specifically, this means that:

\[
\exists x \, P(x) \equiv P(1) \lor P(2) \lor P(3)
\]

Thus, the statement \( \exists x \, P(x) \) is equivalent to saying that **at least one** of \( P(1) \), \( P(2) \), or \( P(3) \) is true.

Would you like a further breakdown or examples of existential quantifiers?

Here are 5 related questions:
1. What is the universal quantifier \( \forall x \, P(x) \) equivalent to in this context?
2. How would you negate the statement \( \exists x \, P(x) \)?
3. What would \( P(x) \) look like if it were a specific proposition (e.g., \( x > 2 \)) in this universe?
4. How does the concept of quantifiers change if the universe \( U \) is infinite?
5. Can there be more than one \( x \) that satisfies \( P(x) \) in the statement \( \exists x \, P(x) \)?

**Tip:** When interpreting statements involving quantifiers, always check the size and elements of the universe to ensure accurate translation.

If the universe of discourse U = {1, 2, 3}, then the statement ∃xP(x) is equivalent to

Explore how the existential quantifier ∃xP(x) applies to the set U = {1, 2, 3}. Learn how to expand the statement to P(1) ∨ P(2) ∨ P(3) and understand the logic behind quantifiers. Suitable for students in Grades 9-12.

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