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Chapter 2: Problem 6

Write the hypothesis and the conclusion of each conditional. \(x=-5\) only if \(x^{2}=25\).

Short Answer

Expert verified
The hypothesis of the conditional statement is \(x^{2}=25\) and the conclusion is \(x=-5\).

Step by step solution

01

Identify the Hypothesis

Firstly, focus on the phrase after 'only if', which represents the hypothesis in a conditional statement. Here, \(x^{2}=25\) is following 'only if', hence it is the hypothesis.
02

Identify the Conclusion

Now focus on the phrase before 'only if', which represents the conclusion of a conditional statement. Considering this, \(x=-5\) is the conclusion.

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Key Concepts

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

Hypothesis
In a conditional statement, the hypothesis is the proposition or statement that is assumed to be true for the sake of the argument. Think of it as the starting point or the given condition. For our example, the phrase after 'only if' helps us identify the hypothesis.

In the statement "\(x=-5\) only if \(x^{2}=25\)," the part \(x^{2}=25\) is the hypothesis. It is the condition under which our initial claim is examined.
Conclusion
The conclusion is the assertion that follows when the hypothesis is true. It represents what results or consequences arise from the initial condition. In the context of conditional statements, it鈥檚 the outcome or result we draw from the hypothesis.

Looking at the example, before the phrase "only if," we see \(x=-5\). This is the conclusion. Therefore, if it is true that \(x^{2}=25\), then the conclusion must follow that \(x=-5\).

When dealing with such statements, it's crucial to interpret the structure correctly to understand what is concluded based on the given hypothesis.
Mathematical Reasoning
Mathematical reasoning involves the steps and logic used to arrive at a conclusion from a given hypothesis. It鈥檚 all about connecting the dots in a logical manner.

In our example, mathematical reasoning requires assessing if the squares of both sides lead to logical truth or consistency.
  • Given the hypothesis \(x^{2} = 25\), logically, \(x\) can be either 5 or -5, since both values when squared equal 25.
  • The conclusion of \(x = -5\) confirms one possibility arising from this condition, demonstrating reasoning through deduction.
Careful scrutiny of the relationship between hypothesis and conclusion is essential for proper reasoning.
Logical Connectors
Logical connectors are terms that link the hypothesis and conclusion to indicate conditions and consequences. Common connectors include "if," "then," "only if," and "if and only if." They serve to establish the logical structure of the statement.

In this exercise, the connector "only if" provides a strict condition; it sets the stage for the relationship.
  • "Only if" suggests that the conclusion requires the hypothesis to be true, but not necessarily the reverse.
  • In our example, \(x=-5\) applies exclusively under the truth of \(x^{2} = 25\).
The right use of logical connectors helps clarify how the parts of a statement interact, essential for correct logical interpretation and mathematical proofs.

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Most popular questions from this chapter

Point \(T\) is the midpoint of \(\overline{R S} . W\) is the midpoint of \(\overline{R T},\) and \(Z\) is the midpoint of \(\overline{W S}\). If the length of \(\overline{T Z}\) is \(x\). find the following lengths in terms of \(x .\) (Hint: Sketch a diagram and let \(y=W T\).) a. \(R W\) b. \(Z S\) c. \(R S\) d. \(W Z\)

Provide a counterexample to show that each statement is false. You may use words or a diagram. If point \(G\) is on \(\overrightarrow{A B},\) then \(G\) is on \(\overrightarrow{B A}\).

Write the hypothesis and the conclusion of each conditional. If \(m \angle 1=90,\) then \(\angle 1\) is a right angle.

Tell whether each statement is true or false. Then write the converse and tell whether it is true or false. \(x=1\) only if \(x^{2}=x\).

Name the definition, postulate, or theorem that justifies the statement about the diagram. If \(D\) is the midpoint of \(\overline{B C},\) then \(B D=\frac{1}{2} B C\)
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