is "3!", the existential quantifier foll DEFINITION: For an open sentence p(x) and a set S, "(! x S) (p(x))" (read "there is a nique x in S such that p(x)") means: (3x ES) (p(x)) and (Vu, vES) ([p(u) and p(v)] ⇒ u = v). a) Explain, in intuitive terms, how "(3! x S) (p(x))" says that there is exactly one ele- ment from S in the truth set of p(x).
is "3!", the existential quantifier foll DEFINITION: For an open sentence p(x) and a set S, "(! x S) (p(x))" (read "there is a nique x in S such that p(x)") means: (3x ES) (p(x)) and (Vu, vES) ([p(u) and p(v)] ⇒ u = v). a) Explain, in intuitive terms, how "(3! x S) (p(x))" says that there is exactly one ele- ment from S in the truth set of p(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question a
![Certainly! Here is the transcription and explanation suitable for an educational website:
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**Unique Existence Quantifier**
There is a third quantifier, called the "unique existence" quantifier, which is used to abbreviate statements like the definition of "function" (Definition 3.14). The symbol for this quantifier is "∃!", the existential quantifier followed by an exclamation point. It is defined as follows.
**Definition:** For an open sentence \( p(x) \) and a set \( S \), "\( (\exists! x \in S) (p(x)) \)" (read "there is a unique \( x \) in \( S \) such that \( p(x) \)") means:
\[
(\exists x \in S) (p(x)) \quad \text{and} \quad (\forall u, v \in S) ([p(u) \quad \text{and} \quad p(v)] \Rightarrow u = v)
\]
a) Explain, in intuitive terms, how "\( (\exists! x \in S) (p(x)) \)" says that there is exactly one element from \( S \) in the truth set of \( p(x) \).
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Transcribed Image Text:Certainly! Here is the transcription and explanation suitable for an educational website:
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**Unique Existence Quantifier**
There is a third quantifier, called the "unique existence" quantifier, which is used to abbreviate statements like the definition of "function" (Definition 3.14). The symbol for this quantifier is "∃!", the existential quantifier followed by an exclamation point. It is defined as follows.
**Definition:** For an open sentence \( p(x) \) and a set \( S \), "\( (\exists! x \in S) (p(x)) \)" (read "there is a unique \( x \) in \( S \) such that \( p(x) \)") means:
\[
(\exists x \in S) (p(x)) \quad \text{and} \quad (\forall u, v \in S) ([p(u) \quad \text{and} \quad p(v)] \Rightarrow u = v)
\]
a) Explain, in intuitive terms, how "\( (\exists! x \in S) (p(x)) \)" says that there is exactly one element from \( S \) in the truth set of \( p(x) \).
---
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