b) Rewrite Definition 3.14 using the unique existential quantifier. c) Suppose f:AB. Rephrase the quantifier statement "(V VE B) (3! u € A) (f(u) = v)" using terminology for functions introduced in this section. d) Express "~(3! x S) (p(x))" without using negation except directly next to the predicate p(x) (careful! this is not easy).
b) Rewrite Definition 3.14 using the unique existential quantifier. c) Suppose f:AB. Rephrase the quantifier statement "(V VE B) (3! u € A) (f(u) = v)" using terminology for functions introduced in this section. d) Express "~(3! x S) (p(x))" without using negation except directly next to the predicate p(x) (careful! this is not easy).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
b, c, d
Transcribed Image Text:**Definition 3.14:**
For any sets \( A \) and \( B \), a function from \( A \) to \( B \) is a set \( S \subseteq A \times B \) which satisfies the following conditions:
a) \( (\forall u \in A)(\exists v \in B)((u, v) \in S) \), and
b) \( (\forall u \in A)(\forall v, w \in B)(((u, v) \in S \land (u, w) \in S) \Rightarrow v = w) \).
The set \( A \) is called the *domain* of the function, and the set \( B \) is called the *co-domain* of the function.
![b) Rewrite Definition 3.14 using the unique existential quantifier.
c) Suppose \( f: A \rightarrow B \). Rephrase the quantifier statement
\[
(\forall v \in B)(\exists u \in A)(f(u) = v)
\]
using terminology for functions introduced in this section.
d) Express "\(\sim(\exists! x \in S)(p(x))\)" without using negation except directly next to the predicate \( p(x) \) (careful! this is not easy).
e) Explain, in intuitive terms, why "\((\exists! x \in S)(p(x))\)" means the same thing as
\[
(\exists x \in S)[p(x) \land (\forall u \in S)[p(u) \implies u = x]]
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc06c0cac-2e9c-4d45-9a45-e5decd8c209b%2F61a15f6c-1dda-4a5d-bf86-fc221922bca7%2F0xdxh1y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:b) Rewrite Definition 3.14 using the unique existential quantifier.
c) Suppose \( f: A \rightarrow B \). Rephrase the quantifier statement
\[
(\forall v \in B)(\exists u \in A)(f(u) = v)
\]
using terminology for functions introduced in this section.
d) Express "\(\sim(\exists! x \in S)(p(x))\)" without using negation except directly next to the predicate \( p(x) \) (careful! this is not easy).
e) Explain, in intuitive terms, why "\((\exists! x \in S)(p(x))\)" means the same thing as
\[
(\exists x \in S)[p(x) \land (\forall u \in S)[p(u) \implies u = x]]
\]
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