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Chapter 1: Problem 4

Use truth tables to verify the associative laws a) \((p \vee q) \vee r \equiv p \vee(q \vee r)\) b) \((p \wedge q) \wedge r \equiv p \wedge(q \wedge r)\)

Short Answer

Expert verified
Both associative laws are verified since the columns are identical in each part.

Step by step solution

01

List all possible truth values

Create a truth table where you list all possible combinations of truth values for the propositions p, q, and r. Since each can be either true (T) or false (F), there will be 8 combinations in total.
02

Calculate \( (p \vee q) \vee r \)

For each row in the truth table, calculate the truth values for \( p \vee q \) first, and then use those values to calculate \( (p \vee q) \vee r \).
03

Calculate \( p \vee (q \vee r) \)

For the same rows, calculate the truth values for \( q \vee r \) first, and then use those values to calculate \( p \vee (q \vee r) \).
04

Compare the results for part (a)

Compare the columns for \( (p \vee q) \vee r \) and \( p \vee (q \vee r) \). If they are identical for all rows, the associative law for disjunction is verified.
05

Calculate \( (p \wedge q) \wedge r \)

For each row, calculate the truth values for \( p \wedge q \) first, and then use those values to calculate \( (p \wedge q) \wedge r \).
06

Calculate \( p \wedge (q \wedge r) \)

For the same rows, calculate the truth values for \( q \wedge r \) first, and then use those values to calculate \( p \wedge (q \wedge r) \).
07

Compare the results for part (b)

Compare the columns for \( (p \wedge q) \wedge r \) and \( p \wedge (q \wedge r) \). If they are identical for all rows, the associative law for conjunction is verified.

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Key Concepts

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

Truth Tables
To understand the associative laws in logic, we start with truth tables. A truth table is a valuable tool for examining all possible truth values of a logical expression.
It lists all combinations of truth values for the involved propositions.
For example, if we have three propositions, p, q, and r, each can be true (T) or false (F). Thus, there are 2x2x2 = 8 possible combinations:
  • TTT
  • TTF
  • TFT
  • TFF
  • FTT
  • FTF
  • FFT
  • FFF

Using truth tables helps us systematically evaluate complex expressions and verify logical identities.
Disjunction (OR)
In logic, the OR operator is known as disjunction, denoted by \(\backslashvee \).
A disjunction expression like \(p \backslashvee q\) is true if at least one of p or q is true.
Let's apply this to verify the associative law using a truth table.
For \( (p \backslashvee q) \backslashvee r \):
  • First, we find \(p \backslashvee q\).
  • Then we use the result to find \((p \backslashvee q) \backslashvee r \).

The same approach is used for \( p \backslashvee (q \backslashvee r)\). If the columns for \((p \backslashvee q) \backslashvee r\) and \( p \backslashvee (q \backslashvee r)\) are identical for all rows, the associative law holds true for disjunction.
Conjunction (AND)
The AND operator in logic is known as conjunction, denoted by \(\backslashwedge \).
A conjunction expression like \( p \backslashwedge q \) is true only if both p and q are true.
To verify the associative law for conjunction using a truth table, we follow these steps:
For \((p \backslashwedge q) \backslashwedge r \):
  • First, find \(p \backslashwedge q\).
  • Then use this result to calculate \((p \backslashwedge q) \backslashwedge r\).

Apply the same process for \( p \backslashwedge (q \backslashwedge r)\).
If the columns for \((p \backslashwedge q) \backslashwedge r\) and \( p \backslashwedge (q \backslashwedge r)\) match for all rows, the associative law for conjunction is verified.
Associative Law Verification
The associative laws in logic state that the way in which logical operations are grouped does not affect their outcome.
For disjunction, the associative law is stated as \((p \backslashvee q) \backslashvee r \backslashequiv p \backslashvee (q \backslashvee r)\).
For conjunction, it is \((p \backslashwedge q) \backslashwedge r \backslashequiv p \backslashwedge (q \backslashwedge r)\).
  • To verify these laws, we calculate the expressions for different groupings.
  • We use truth tables to confirm that both sides of the associative law yield the same results for all possible truth values.

If the results match consistently, the associative law is confirmed. This means that for disjunction and conjunction, how you group propositions in logic does not change the truth value.

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