Question

Give an example of a universal set \(U\) and predicates \(P\) and \(Q\) such that \((\forall x P(x)) \rightarrow\) \((\forall x Q(x))\) is true but \(\forall x(P(x) \rightarrow Q(x))\) is false.

Short answer

The universal set is \(U = \{1, 2, 3\}\), with predicates \(P(x)\) as "\(x\) is odd" and \(Q(x)\) as "\(x = 1\)."

Step-by-step solution

Define the Universal Set

Let's define a universal set \(U = \{1, 2, 3\}\). This set contains three elements that we'll use to evaluate predicates \(P\) and \(Q\).

Define Predicates P and Q

Define predicate \(P(x)\) as "\(x\) is an odd number." Thus, \(P(1)\) is true, \(P(2)\) is false, and \(P(3)\) is true. Now, define predicate \(Q(x)\) as "\(x = 1\)." Consequently, \(Q(1)\) is true, \(Q(2)\) is false, and \(Q(3)\) is false.

Evaluate \((\forall x P(x)) \rightarrow (\forall x Q(x))\)

\(\forall x P(x)\) is false because \(P(2)\) is false. In logical implication \(A \rightarrow B\), if \(A\) is false, \(A \rightarrow B\) is automatically true, regardless of the truth value of \(B\). Therefore, \((\forall x P(x)) \rightarrow (\forall x Q(x))\) is true.

Evaluate \(\forall x (P(x) \rightarrow Q(x))\)

We need to check each individual element in \(U\):\- \(P(1) \rightarrow Q(1)\) is true since both are true,\- \(P(2) \rightarrow Q(2)\) is true since \(P(2)\) is false,\- \(P(3) \rightarrow Q(3)\) is false since \(P(3)\) is true but \(Q(3)\) is false. \Hence, \(\forall x (P(x) \rightarrow Q(x))\) is false as one of the conditions, specifically \(x = 3\), fails.

Key concepts

Universal Set

In predicate logic, a universal set, denoted as \( U \), is the collection of all possible elements under consideration for a given discussion or problem. It's akin to a universe where every potential subject for your predicates resides.

For example, if we choose our universal set \( U = \{1, 2, 3\} \), it means that we will only be discussing numbers within this set. Each element in this universal set can be assessed using logical rules or predicates that we define. By clearly defining the universal set, we establish the boundary within which our logical statements and evaluations will take place.

By limiting our context to \( \{1, 2, 3\} \), we break down complex logical evaluations into manageable tasks by ensuring we're only considering these elements when testing predicates like \( P(x) \) or \( Q(x) \). This allows for precise and clear assessments of logical expressions. It's crucial to understand what elements our universal set includes to accurately interpret predicates and their logical implications.

Logical Implications

Logical implications are statements that express a condition of 'if...then...' between two logical assertions. In logical terms, an implication \( A \rightarrow B \) states "If \( A \) is true, then \( B \) must also be true."

This kind of statement has a fascinating rule: whenever \( A \) is false, \( A \rightarrow B \) is automatically true, regardless of whether \( B \) is true or false. This rule aligns with formal logic principles where a false premise can lead to any conclusion.

In our given exercise, we observe this in the implications \( (\forall x P(x)) \rightarrow (\forall x Q(x)) \). Since \( \forall x P(x) \) is false due to \( P(2) \) being false, the entire implication holds true. Understanding the mechanics of logical implications helps in reasoning about complex predicate logic problems, ensuring that each step follows logically from the previous conditions, thus maintaining integrity in logical reasoning.

Truth Values

In logic, each statement or predicate is assigned a truth value: either true or false. Truth values are pivotal in evaluating the validity of logical expressions and helping us reason through complex predicate logic.

For example, consider predicates \( P(x) \) and \( Q(x) \) with assigned values based on conditions: \( P(x) \) being true if \( x \) is odd and \( Q(x) \) true if \( x = 1 \). For each element in \( U \), we determine whether predicates hold true or false:

• \( P(1) \) and \( Q(1) \) are both true.
• \( P(2) \) is false and \( Q(2) \) is false.
• \( P(3) \) is true, but \( Q(3) \) is false.

These truth values help us assess the entire logical statement \( \forall x (P(x) \rightarrow Q(x)) \), where each specific outcome affects whether the overall statement is true or false. For logical expressions to be fully understood, each part must be assessed for its truth value, as seen when \( P(3) \) failing to imply \( Q(3) \) renders the whole statement false.

Understanding truth values solidifies one’s grasp of logical arguments, making it easier to predict outcomes and validate logical expressions in predicate logic scenarios.