Let f be a real valued real function. We say that f is increasing if a < b ⇒ f(a) ≤ f(b). a) Write out this definition in terms of a quantified propositional function. b) Write out... Let f be a real valued real function. We say that f is increasing if a < b ⇒ f(a) ≤ f(b). a) Write out this definition in terms of a quantified propositional function. b) Write out the negation of this definition. c) Write out the definition of a decreasing function by flipping the inequality used in defining an increasing function.
Understand the Problem
The question is asking for a formal expression of the definition of an increasing function using quantified propositional functions, followed by the negation of that definition, and a similar definition for a decreasing function by inverting the inequality. This involves translating mathematical concepts into logical and mathematical notation.
Answer
Increasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \leq f(x_2)) $$ Negation: $$ \exists x_1, x_2 \in \text{domain}, (x_1 < x_2 \land f(x_1) > f(x_2)) $$ Decreasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \geq f(x_2)) $$
Answer for screen readers
The definitions and negations are:
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Increasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \leq f(x_2)) $$
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Negation of Increasing Function: $$ \exists x_1, x_2 \in \text{domain}, (x_1 < x_2 \land f(x_1) > f(x_2)) $$
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Decreasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \geq f(x_2)) $$
Steps to Solve
- Definition of an Increasing Function
An increasing function is one where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \leq f(x_2)$.
This can be expressed using quantified propositional functions as:
$$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \leq f(x_2)) $$
- Negation of the Definition of an Increasing Function
To find the negation, we need to flip the implication and the inequality:
Negation states that there exists at least one pair $(x_1, x_2)$ where $x_1 < x_2$ but $f(x_1) > f(x_2)$.
It can be expressed as:
$$ \exists x_1, x_2 \in \text{domain}, (x_1 < x_2 \land f(x_1) > f(x_2)) $$
- Definition of a Decreasing Function
A decreasing function is defined as one where, for any two points $x_1$ and $x_2$ in its domain, if $x_1 < x_2$, then $f(x_1) \geq f(x_2)$.
This can be represented with quantified propositional functions as:
$$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \geq f(x_2)) $$
The definitions and negations are:
-
Increasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \leq f(x_2)) $$
-
Negation of Increasing Function: $$ \exists x_1, x_2 \in \text{domain}, (x_1 < x_2 \land f(x_1) > f(x_2)) $$
-
Decreasing Function: $$ \forall x_1, x_2 \in \text{domain}, (x_1 < x_2 \implies f(x_1) \geq f(x_2)) $$
More Information
In mathematics, functions can be classified as increasing or decreasing based on how their output values change relative to their input values. The concept of monotonicity is crucial in calculus and analysis, providing foundational understandings in various mathematical studies.
Tips
- Confusing increasing and decreasing functions: Be careful to check the direction of the inequalities.
- Misinterpreting the negation of statements: Make sure to correctly translate implications and equate inequalities when finding negations.
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