Problem 1 - A Gentle Introduction to the Art of Mathematics

Question

There is a common variant of the existential quantifier, \(\exists !,\) if you write \(\exists ! x, P(x)\) you are asserting that there is a unique element in the universe that makes \(P(x)\) true. Determine how to negate the sentence \(\exists ! x, P(x)\)

Step-by-Step Solution

Answer

\(\forall x, eg P(x) \vee \exists y, (P(y) \wedge y eq x)\)

Step 1: Understand the original statement

The statement \(\exists ! x, P(x)\) asserts that there is exactly one element in the universe that makes \(P(x)\) true.

Step 2: Translate the existential uniqueness

The statement \(\exists ! x, P(x)\) can be equivalently written as \(\exists x, P(x) \wedge \forall y, (P(y) \rightarrow y=x)\). This means there is at least one element \(x\) such that \(P(x)\) is true, and for every \(y\), if \(P(y)\) is true, then \(y\) must be identical to \(x\).

Step 3: Negate the translated statement

To negate \(\exists ! x, P(x)\), negate the equivalent form: \(eg (\exists x, P(x) \wedge \forall y, (P(y) \rightarrow y=x))\).

Step 4: Apply De Morgan's laws

Using De Morgan's laws, the negation becomes \(eg (\exists x, P(x)) \vee eg (\forall y, (P(y) \rightarrow y=x))\).

Step 5: Distribute the negations

The statement \(eg (\exists x, P(x))\) is equivalent to \(\forall x, eg P(x)\), and \(eg (\forall y, (P(y) \rightarrow y=x))\) is equivalent to \(\exists y, eg (P(y) \rightarrow y=x)\).

Step 6: Simplify the implication

The negation \(eg (P(y) \rightarrow y=x)\) is equivalent to \(P(y) \wedge y eq x\).

Step 7: Combine the results

Substituting back, the negation of \(\exists ! x, P(x)\) is \(\forall x, eg P(x) \vee \exists y, (P(y) \wedge y eq x)\).

Key Concepts

existential quantifier

The existential quantifier, denoted by \(\exists\), is employed in mathematical logic to express that there is at least one element in a given set that satisfies a particular property. For instance, the statement\(\exists x, P(x)\) asserts that there is at least one element \x\ in the universe of discourse for which the property \P(x)\ holds true.
An example can help illustrate this better: Consider the statement, 'There exists a number x such that x + 2 = 5'. This translates to \(\exists x, x + 2 = 5\). Here, the number 3 satisfies the condition because 3 + 2 equals 5.
The key points to remember about the existential quantifier are:

• It asserts the existence of at least one element that satisfies the property.
• It is represented by the symbol \(\exists\).
• The negation of \(\exists x, P(x)\) is \(\forall x, \eg P(x)\), meaning 'for all x, the property P(x) does not hold'.

uniqueness

Uniqueness in mathematical logic asserts that there is one and only one element that satisfies a specified property. The notation \(\exists ! x, P(x)\) is used to indicate this 'unique existence'.
When we write \(\exists ! x, P(x)\), we mean there is a unique element x such that P(x) is true. In other words, not only does such an element exist, but it is unique - no other element satisfies the property.
To formalize this, the statement \(\exists ! x, P(x)\) can be rewritten using standard quantifiers as follows:
\[\exists x (P(x) \wedge \forall y (P(y) \rightarrow y = x))\]
This expression states that there exists one x for which P(x) is true and for any y, if P(y) is true, then y must be the same as x.
Understanding uniqueness is crucial because it ensures whether a solution to a problem is one of a kind. For instance, in the mathematical context, if a solution is unique, then there is no ambiguity or possibility for multiple answers.

negation

Negation is a fundamental concept in logic that involves flipping the truth value of a statement. If a statement is true, its negation is false and vice versa. For example, the negation of the statement 'It is raining' would be 'It is not raining'.
In the context of logical quantifiers, negating a statement involves understanding how to apply De Morgan's laws. These laws pertain to the negation of conjunctions and disjunctions and are crucial in simplifying logical expressions.
Let's look at negating \(\exists ! x, P(x)\), which asserts that there exists a unique x such that P(x) is true.
Following the steps:

• Rewrite the statement using standard quantifiers: \(\exists x, P(x) \wedge \forall y, (P(y) \rightarrow y = x)\).
• Apply negation to the entire expression: \(\eg (\exists x, P(x) \wedge \forall y, (P(y) \rightarrow y = x))\).
• Use De Morgan's laws to distribute the negation: \(\eg (\exists x, P(x)) \vee \eg (\forall y, (P(y) \rightarrow y = x))\).
• Interpret each part: \(\eg (\exists x, P(x))\) becomes \(\forall x, \eg P(x)\), and \(\eg (\forall y, (P(y) \rightarrow y = x))\) becomes \(\exists y, \eg (P(y) \rightarrow y = x)\).
• Finally, simplify the implication: \(\eg (P(y) \rightarrow y = x)\) is equivalent to \(P(y) \wedge y \eq x\).

This results in the final negation: \(\forall x, \eg P(x) \vee \exists y, (P(y) \wedge y \eq x)\).
This means either all x do not satisfy P(x), or there is some y such that P(y) is true, but y is not identical to x.