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Chapter 1: Problem 5

Find the truth value of each compound statement. $$(5<8) \text { and }(2+3=4)$$

Short Answer

Expert verified
The truth values of the given compound statements are: \((5<8)\) is True, and \((2+3=4)\) is False. So, the truth value of the compound statement, using the "and" operator, is False.

Step by step solution

01

Evaluate the first statement \((5

The first statement is an inequality. To evaluate this, we need to check if 5 is indeed less than 8. Since 5 is indeed less than 8, the statement \((5<8)\) is True.
02

Evaluate the second statement \((2+3=4)\)

The second statement is an equation. To evaluate this, we must check if the sum of 2 and 3 is equal to 4. 2 + 3 = 5, not 4. Thus, the statement \((2+3=4)\) is False.
03

Combine the truth values with the "and" operator

Now that we have found the truth value for each statement, we'll use the "and" operator to combine them. In logic, the "and" operator returns True only if both statements are True. In our case, one statement is True (\((5<8)\)) and the other statement is False (\((2+3=4)\)). Therefore, the truth value of the compound statement is False.

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Key Concepts

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

Truth Value
In logic, each statement can either be true or false, which we refer to as its truth value. To determine the truth value, we evaluate the statement under consideration. For example, consider the statement "5 is less than 8." By checking this statement, we find that since 5 is indeed less than 8, its truth value is "True."
However, if we take another statement such as "2 plus 3 equals 4," we find it false because the sum is 5, not 4. So, its truth value is "False."
This concept is crucial in determining the validity of logical statements and arguments, as it provides the basis for evaluating logical expressions and deriving conclusions.
Inequality
Inequalities express a relationship where one side is either less than or greater than the other. Symbols like "<" (less than) or ">" (greater than) are used to define inequalities.
For example, the inequality \( 5 < 8 \) means that 5 is less than 8, which is a true statement.
By comparing the values involved in an inequality, we can determine if the statement is true or false. This contributes to the truth value when involved in logical expressions.
Compound Statement
A compound statement brings together two or more individual statements, often joined by logical operators, to form a new statement. These statements can become complex depending on how many propositions are combined.
For example, consider "\( (5 < 8) \) and \( (2 + 3 = 4) \)." This is a compound statement that uses the "and" operator to connect two simpler statements.
Its truth value is determined by the truth values of its parts and the logical operator connecting them. Understanding compound statements helps in building complex logical expressions and solving more advanced problems.
Logical Operators
Logical operators are symbols or words used to connect individual statements into compound statements and to determine their truth values. Common logical operators include "and," "or," and "not."
The "and" operator requires all combined statements to be true for the compound statement to be true. If any part is false, as in our example with "\( (5 < 8) \) and \( (2 + 3 = 4) \)," the "and" operator results in a false truth value.
Understanding how logical operators work is essential for evaluating logical statements and solving exercises involving compound expressions in discrete mathematics.

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