Question

For each quantified formula that follows: find a universe \(U\) and predicates \(A\) and \(B\) in which the formula is true and \(U, A\) and \(B\) in which it is false. (a) \(\forall x(((A(x) \vee B(x)) \wedge \neg(A(x) \wedge B(x)))\) (b) \(\forall x \forall y(P(x, y) \rightarrow P(y, x))\) (c) \(\forall x(P(x) \rightarrow \exists y Q(x, y))\) (d) \(\exists x(A(x) \wedge \forall y B(x, y))\) (e) \(\forall x A(x) \rightarrow(\forall x B(x) \rightarrow(\forall x(A(x) \rightarrow B(x))))\)

Short answer

Each formula can have true and false scenarios by choosing appropriate predicates and universe.

Step-by-step solution

Understanding the Problem

We need to evaluate each quantified logic formula (a) to (e) by finding a universe \(U\) and predicates \(A\) and \(B\) such that the formula holds true and another scenario where it does not. We will break each formula into logical components and explore possible universes and predicates.

Formula (a) Analysis

The formula is \( \forall x(((A(x) \vee B(x)) \wedge eg(A(x) \wedge B(x)))) \). Consider \( U = \{1, 2\} \), with \( A(x) \) as \( x = 1 \) and \( B(x) \) as \( x = 2 \). The formula is true as there is no overlap between \( A(x) \) and \( B(x) \). However, redefine \( A(x) \) as true for both 1 and 2, this formula with \( B(x) \) also true makes it false due to overlap.

Formula (b) Analysis

The formula is \( \forall x \forall y(P(x, y) \rightarrow P(y, x)) \). Let \( U = \{1\} \) and \( P(x, y) \) be true for all combinations. The formula is trivially true. False scenario: Let \( U = \{1, 2\} \), \( P(1, 2) = \text{true}\) and \( P(2, 1) = \text{false} \). This setup makes the formula false.

Formula (c) Analysis

The formula is \( \forall x(P(x) \rightarrow \exists y Q(x, y)) \). Consider \( U = \{1, 2\} \) with \( P(x)\) true for all \( x \), and \( Q(x, y) \) true if \( x = y \), making the formula true. Now, let \( U \) be the same but define \( P(1) = \text{true}\) and \( Q(1, y)\) as \( \text{false}\) for all \( y \). This makes the formula false for \( x = 1\).

Formula (d) Analysis

The formula is \( \exists x(A(x) \wedge \forall y B(x, y)) \). Let \( U = \{1\} \), \( A(1) = \text{true}\), and \( B(1, 1) = \text{true}\). The formula is true. A false universe is \( U = \{1, 2\} \), where \( A(1) = \text{true}\), and \( B(1, y) = \text{true}\) only for \( y = 1 \). The failure to satisfy all \( y \) in \( U \) makes the formula false.

Formula (e) Analysis

The formula is \( \forall x A(x) \rightarrow(\forall x B(x) \rightarrow(\forall x(A(x) \rightarrow B(x)))) \). With \( U = \{1\} \), \( A(1) = \text{true}\), and \( B(1) = \text{true}\), this is true. For a false outcome, let \( U = \{1, 2\}\), \( A(x)\) true for all \( x \), but \( B(2) = \text{false}\), disrupting \( \forall x(A(x) \rightarrow B(x))\).

Key concepts

Predicates

In the realm of logical expressions, predicates are essential building blocks. A predicate is essentially a statement or proposition that may depend on a variable. For example, in the expression \(A(x)\), \(A\) is a predicate that could mean "\(x\) is an even number." The value of \(x\) determines whether the statement is true or false.

Predicates allow us to express properties about things or individuals from a specific collection, often known as the universe of discourse or domain. They can be as simple as one variable, such as \(A(x)\), or they might involve multiple variables, like \(P(x, y)\).

Understanding predicates is vital for evaluating logical statements because they enable the specification of conditions that elements of the universe must satisfy. This framework allows us to explore whether complex logical formulas hold true under different conditions.

Universe in Logic

The universe in logic refers to the set of all possible elements that you are considering in a particular logical context. It is also known as the domain of discourse. This set is essential, as it defines the range over which a predicate is evaluated.

For instance, if we're considering the universe \(U = \{1, 2, 3\}\), any predicate involving a variable, say \(A(x)\), will consider each element in \(U\). Depending on the context, universes can be finite, like \(\{1, 2, 3\}\), or infinite, such as the set of all real numbers.

The size and elements of the universe significantly impact whether a quantified statement is true or false. For example, if the universe changes, the truth of the statement \(\forall x A(x)\) might change as well. Therefore, understanding the specified universe is crucial when analyzing logical statements.

Truth Values in Logic

Truth values in logic reflect whether a particular statement or proposition is true or false. In mathematical logic, these values are typically limited to true and false, represented precisely as '1' and '0'.

When it comes to predicate logic, assigning truth values involves considering the predicates within the expression on an element-by-element basis from the universe. For example, \(A(x)\) might be true for some \(x\) and false for others, depending on the properties defined by \(A\).

A logical formula's overall truth depends on the compounded truth values of its components. In formulas like \(\forall x P(x)\), the statement is true if \(P(x)\) holds for every \(x\) in the universe. If even one element does not satisfy \(P(x)\), the formula is false. Hence, understanding how truth values are derived is pivotal in predicate logic.

Predicate Logic Analysis

Predicate logic analysis involves examining logical statements to determine their validity under different conditions, such as varied predicates or universes. It often deals with quantified formulas involving "for all" (\(\forall\)) or "there exists" (\(\exists\)), and predicates that rely on one or more variables.

For instance, consider the expression \(\forall x (A(x) \vee B(x))\). Analyzing this involves checking if every element \(x\) in the universe \(U\) makes the statement true. This requires evaluating both \(A(x)\) and \(B(x)\). If at least one of these is true for each \(x\), the whole expression holds.

Predicate logic analysis is essential for solving problems in logic because it systematically dissects statements into their logical components, allowing you to see where they hold true and under what conditions they fail. By developing this analytical skill, you'll be better equipped to handle complex logical expressions in various contexts.