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Chapter 1: Problem 38

Determine the truth value of each proposition. \(3+5<2\) if and only if \(1+3=4\).

Short Answer

Expert verified
The truth value of the proposition '3+5<2 if and only if 1+3=4' is false.

Step by step solution

01

Determine the Truth Value of the First Proposition

Calculate the sum and compare with 2. The sum of 3 and 5 is 8. Comparing this with 2, the inequality 8<2, it is clearly false since 8 is not less than 2.
02

Determine the Truth Value of the Second Proposition

Calculate the sum and compare with 4. The sum of 1 and 3 is 4. Comparing this with 4, the equality 4=4, it is clearly true since 4 does equal 4.
03

Determine the Truth Value of the Compound Proposition

With 'if and only if', the compound proposition is true only if both the individual propositions are true or both are false. In this case, the first proposition is false and the second is true, therefore the entire compound proposition '3+5<2 if and only if 1+3=4' is false.

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Key Concepts

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

Truth Value
Truth values are fundamental to logic. They determine whether a statement or proposition is true or false.
The simplest propositions are evaluated based on whether they correspond to reality or meet certain conditions.
For example, the proposition "3+5<2" does not hold true in reality.
This makes it false, thereby having a truth value of 'false'. Similarly, simple arithmetic propositions like "1+3=4" are true if the arithmetic is correct.
Thus, the truth value of the second proposition is 'true'. Instead of dealing with just true or false, it also helps in analyzing complex logical statements by examining each individual part.
Inequality
An inequality is a mathematical statement showing that two values are not equal and have a different relationship such as greater than or less than.
In our exercise, "3+5<2" is given as an inequality proposition.
Here, when we calculate 3+5, we get 8. It becomes clear that 8 is not less than 2, confirming that the inequality is false.
This reflects not only an incorrect mathematical logic but also an opportunity to practice testing the truth of such statements through calculations.
  • Inequalities often use symbols such as "<", "<=" for less than or equal,
    and ">", ">=" for greater than or equal.
  • Understanding inequalities is key to evaluating logical and mathematical propositions correctly.
Equality
Equality in mathematics is an assertion that two expressions represent the same value.
For example, in the proposition "1+3=4", the equality sign "=" is used to affirm that the sum on one side is equal to the number on the other side.
Equality propositions are vital in determining the correctness of equations.
When evaluating the expression "1+3=4", we find it true as both sides of the equality represent the number 4.
This truth allows us to assert the validity of this proposition.
Compound Propositions
Compound propositions are statements formed by combining two or more simple propositions using logical connectives.
Common connectives include 'and', 'or', and 'if and only if'.
In our practice, the compound proposition is "3+5<2 if and only if 1+3=4".
  • The term 'if and only if' indicates a biconditional proposition meaning both component propositions must be true, or both false, for the entire statement to be true.
  • The outcome here is false because the first proposition is false while the second one is true.
  • Understanding compound propositions helps in complex problem-solving across various logical contexts, requiring each part's truth value to be accurately determined.

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Most popular questions from this chapter

Let \(P\) denote the set of integers greater than \(1 .\) For \(i \geq 2,\) define $$X_{i}=\\{i k \mid k \in P\\}$$ Describe \(P-\bigcup_{i=2}^{\infty} X_{i}\).

Answer true or false. $$ \\{x\\} \in\\{x\\} $$

Assume that \(\forall x \forall y P(x, y)\) is false and that the domain of discourse is nonempty. Which of must also be false? Prove your answer. $$ \exists x \exists y P(x, y) $$

Describe the symmetric difference of sets \(A\) and \(B\) in words.

Determine the truth value of each statement. The domain of discourse is \(\mathbf{R}\). Justify your answers. $$ \exists x\left(x>1 \rightarrow x^{2}>x\right) $$
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