Problem 68 - Discrete Mathematics with Applications

Question

Define the quantifier \(\exists !\) in terms of the quantifiers \(\exists\) and \(\forall\).

Step-by-Step Solution

Answer

The unique existential quantifier \((\exists !x) P(x)\) can be defined in terms of the existential and universal quantifiers as follows: \((\exists x)(P(x) \land (\forall y)(P(y) \Rightarrow (y = x)))\). This definition asserts that there exists an object \(x\) with the desired property \(P(x)\) and that any other object \(y\) with the property \(P(y)\) must be equal to \(x\), ensuring uniqueness.

Step 1: Understanding the unique existential quantifier (\(\exists !\))

The unique existential quantifier (\(\exists !\)) asserts that there is exactly one object with a certain property. We can use the other quantifiers, "existential" (\((∃x)\), which means "there exists an x") and "universal" (\((∀x)\), which means "for all x"), to express the definition of the unique existential quantifier.

Step 2: Defining uniqueness using the existential and universal quantifiers

To define the unique existential quantifier, we need to express two conditions: 1. There exists an object with the desired property. 2. There are no other objects with the desired property. Let \(P(x)\) be a property of the individual \(x\).

Step 3: Express the first condition

To express the first condition, we simply use the existential quantifier. We need to say that there exists an object \(x\) with the property \(P(x)\): \[ (∃x) P(x) \]

Step 4: Express the second condition

To express the second condition, we need to say that if \(y\) also has the property \(P(y)\), then \(y\) must be equal to \(x\). We use the universal quantifier to consider all possible values of \(y\): \[ (∀y) (P(y) \Rightarrow (y = x)) \] Here, \(P(y) \Rightarrow (y = x))\) means '"if \(y\) has property \(P(y)\), then \(y\) must equal \(x\)" (which ensures uniqueness).

Step 5: Combining both conditions to define the unique existential quantifier

Finally, we combine both conditions (from Steps 3 and 4) to define the unique existential quantifier \(\exists !\): \[ (∃!x) P(x) \equiv ( ∃x) (P(x) \land (∀y) (P(y) \Rightarrow (y = x))) \] Thus, the quantifier \(\exists !\) is defined in terms of the quantifiers \(\exists\) and \(\forall\).

Key Concepts

Understanding the Existential Quantifier

The existential quantifier is a logical notation used in mathematics and philosophy. It is represented by the symbol \( \exists \). When we see the existential quantifier \((\exists x)\), it signifies that in a given set or domain, there is at least one element, \(x\), that satisfies a particular property or condition. For instance, if we have a property \(P(x)\), the expression \((\exists x) P(x)\) translates to "There exists at least one \(x\) such that \(P(x)\) is true."

The existential quantifier is crucial in many mathematical and logical expressions because it helps to assert the existence of an element meeting a particular criterion. It does not provide details about how many of such elements exist, only that there is at least one. This is a foundational element in constructing and understanding logical statements and proofs.

Exploring the Universal Quantifier

The universal quantifier is another fundamental component of logical expressions, symbolized by \( \forall \). It denotes that a particular property or condition holds true for all elements in a given set or domain. When you encounter an expression like \((\forall x) P(x)\), it means "For every element \(x\), the property \(P(x)\) is true."

The universal quantifier is extremely useful in proving statements that apply universally, such as mathematical theorems. For example, in mathematics, stating that "For every number \(x\), the sum of \(x\) and 0 is \(x\)" is expressed logically using the universal quantifier. This approach ensures that a property is not just true for some or most elements, but for all possible elements in the discussion.

• Applies globally within a given scope or boundary
• Essential in formulating theorems and proofs

Crafting Logical Expressions

Logical expressions are combinations of symbols and operators that represent statements or propositions in a logical form. These expressions are constructed using quantifiers like the existential (\( \exists \)) and universal (\( \forall \)) quantifiers, along with logical connectors such as AND (\( \land \)), OR (\( \lor \)), NOT (\( eg \)), and implication (\( \Rightarrow \)).

In crafting logical expressions, clarity and preciseness are key. For example, the expression \((\exists x)(P(x) \land (\forall y)(P(y) \Rightarrow y = x))\) is a logical expression defining the unique existential quantifier. This expression states that there is exactly one \(x\) such that the property \(P(x)\) is true, reflecting a combination of existence and uniqueness. Logical expressions allow us to articulate detailed relationships and reason rigorously about potentially complex scenarios.

• Employs different quantifiers and logical connectors
• Facilitates the expression of complex ideas and relationships
• Core in developing rigorous logical proofs