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Chapter 10: Problem 20

Let \(A\) be the set of all statement forms in three variables \(p, q\), and \(r . \mathscr{R}\) is the relation defined on \(A\) as follows: For all \(P\) and \(Q\) in \(A\), \(P \mathscr{R} Q \Leftrightarrow P\) and \(Q\) have the same truth table.

Short Answer

Expert verified
To determine if two statement forms \(P\) and \(Q\) are related under the relation \(\mathscr{R}\) on set \(A\), which consists of all statement forms in three variables \(p, q, r\), we need to check if their truth tables are the same. First, create the truth tables for each statement form \(P\) and \(Q\), with \(8\) rows representing all possible combinations of truth values for the variables. Then, compare the resulting truth values of \(P\) and \(Q\) for each row in their respective truth tables. If the truth values match for all rows, then \(P\) and \(Q\) have the same truth table and \(P \mathscr{R} Q\).

Step by step solution

01

Define set A and relation \(\mathscr{R}\)

The set \(A\) consists of all statement forms in three variables: \(p\), \(q\), and \(r\). The relation \(\mathscr{R}\) is defined as \(P\) \(\mathscr{R}\) \(Q\) if and only if \(P\) and \(Q\) have the same truth table. Some examples of elements in set \(A\) include: \(p \land q\), \(p \lor q\), \((p \to q) \land r\), \((p \leftrightarrow q) \lor (\lnot r)\), and so on. To check if two statement forms \(P\) and \(Q\) are related under \(\mathscr{R}\), we need to verify if their truth tables are the same.
02

Create truth tables for statement forms P and Q

Start by creating the truth tables for each statement form \(P\) and \(Q\). A truth table for a statement form with three variables will have \(2^3 = 8\) rows, representing all possible combinations of truth values for the variables \(p, q, r\). For each row, compute the resulting truth value of the statement forms \(P\) and \(Q\), based on the truth values of the three variables.
03

Compare the truth tables of statement forms P and Q

To determine if \(P\) and \(Q\) are related under \(\mathscr{R}\), we need to compare the resulting truth values of \(P\) and \(Q\) for each of the \(8\) rows in their respective truth tables. If the truth values match for each row, then \(P\) and \(Q\) have the same truth table and are related under \(\mathscr{R}\).
04

Example: Comparing statement forms P and Q

Consider the following example of two statement forms: \(P = p \land q\) and \(Q = p \land (\lnot q \lor q)\). To check if \(P \mathscr{R} Q\), create the truth tables for both statement forms \(P\) and \(Q\). After completing the truth tables, compare the resulting truth values for each row. In this case, the truth values for each row match, so \(P\) and \(Q\) have the same truth table and \(P \mathscr{R} Q\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Truth Tables
Truth tables are essential tools in discrete mathematics, especially when dealing with logical statements. They help us visualize the truth values of particular statements based on different combinations of truth values of involved variables. For statement forms involving three variables鈥攍ike in this exercise where the variables are \( p, q, \) and \( r \)鈥攁 truth table will consist of \( 2^3 \) or 8 different rows.
Each row represents a specific combination of truth values (True or False) for the variables.
  • The first row might represent all variables being True: \( p = T, q = T, r = T \).
  • The last row would be all False: \( p = F, q = F, r = F \).
To determine if two statements, say \( P \) and \( Q \), are equivalent, we construct their respective truth tables. If for every possible combination of the truth values of \( p, q, \) and \( r \), both statements have the same truth values, the truth tables are identical. This identity of truth tables indicates logical equivalence between the statements.
Logical Equivalence
Logical equivalence plays a pivotal role in understanding relationships between different statement forms in discrete mathematics. Two statement forms \( P \) and \( Q \) are logically equivalent if they yield identical truth values across all possible interpretations of their variables, which means their truth tables must match completely.
This relation is denoted as \( P \equiv Q \).

The logical equivalence tells us that the truth of one statement guarantees the truth of another, no matter how the individual variables are interpreted.
When comparing statements like \( p \land q \) with \( p \land (\lnot q \lor q) \), you construct the truth tables and find their truth outputs are the same across all rows. Thus, these statements are logically equivalent because they express the same logical idea, albeit through different syntactic structures.
Statement Forms
In discrete mathematics, statement forms are expressions composed of variables and logical connectives that can take on various truth values. Each form represents a different logical statement, and they vary based on the logical operations used and the structure of the expression.
  • For instance, simple forms like \( p \land q \) combine variables using conjunction.
  • More complex forms, such as \( (p \leftrightarrow q) \lor (\lnot r) \), use a combination of logical connectives.
Statement forms help us explore how different logical operators affect the truth value of expressions. They also serve a primary role when verifying logical equivalences since constructing their truth tables lets us compare different forms to see if they essentially convey the same meaning.
By doing so, statement forms become the building blocks in arguments, proofs, and other logical analyses, allowing us to break down complex ideas into manageable parts and understand their logical implications.

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Most popular questions from this chapter

Define a relation \(R\) from \(\mathbf{R}\) to \(\mathbf{R}\) as follows: For all \((x, y) \in \mathbf{R} \times \mathbf{R}, \quad x R y \Leftrightarrow y=\lfloor x\rfloor .\) Draw the graphs of \(R\) and \(R^{-1}\) in the Cartesian plane.

Let \(S\) be the set of all strings of \(a\) 's and \(b\) 's. Define a relation \(R\) on \(S\) as follows: For all \(t \in S\), $$ s R t \quad \Leftrightarrow \quad l(s) \leq l(t) $$

Let \(A=\\{4,5,6\\}\) and \(B=\\{5,6,7\\}\) and define binary relations \(R, S\), and \(T\) from \(A\) to \(B\) as follows: For all \((x, y) \in A \times B, \quad(x, y) \in R \quad \Leftrightarrow \quad x \geq y .\) For all \((x, y) \in A \times B, \quad x S y \Leftrightarrow 2 \mid(x-y)\). \(T=\\{(4,7),(6,5),(6,7)\\} .\) a. Draw arrow diagrams for \(R, S\), and \(T\). b. Indicate whether any of the relations \(R, S\), and \(T\) are functions.

Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. Let \(A\) be a nonempty set and \(\mathscr{P}(A)\) the power set of \(A\). Define the "relative complement" relation \(\mathscr{C}\) on \(\mathscr{P}(A)\) as follows: For all \(X, Y \in \mathscr{P}(A), X \mathscr{E} Y \Leftrightarrow Y=A-X\).

Suppose \(R\) and \(S\) are binary relations on a set \(A\). a. If \(R\) and \(S\) are reflexive, is \(R \cap S\) reflexive? Why? \(\boldsymbol{H}\) b. If \(R\) and \(S\) are symmetric, is \(R \cap S\) symmetric? Why? c. If \(R\) and \(S\) are transitive, is \(R \cap S\) transitive? Why?
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