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Hypotheses and Conclusions

A free Logic lesson from the “Conditionals” unit, with a worked example and practice problems including step-by-step solutions.

The hypothesis is the condition being assumed; the conclusion is what follows from that condition. Learning objective: Identify the if-part and then-part of a conditional. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: In 'If a number is divisible by 4, then it is even,' the hypothesis is 'divisible by 4.' Example 2: The contrapositive is 'If a number is not even, then it is not divisible by 4.' A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: If-then reasoning is the language of theorems, algebra rules, geometry proofs, and programming branches.

Worked example

Problem. Example case A (Hypotheses and Conclusions): Worked example: Which rule or habit best matches Hypotheses and Conclusions?

  1. Hypotheses and Conclusions targets a specific reasoning habit.
  2. Hypotheses and Conclusions focuses on identify the if-part and then-part of a conditional.
  3. The other choices either overclaim or change the logical relationship.

Answer: Hypotheses and Conclusions focuses on identify the if-part and then-part of a conditional.

Practice problems

1. Practice case A (Hypotheses and Conclusions): In "If a number is divisible by 4, then the number is even," what is the hypothesis?

Choices: a number is divisible by 4 · the number is even · if · then

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Hypotheses and Conclusions): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
  2. Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  3. The hypothesis is the if-part.
  4. The conclusion is the then-part.
  5. Here the hypothesis is a number is divisible by 4.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: a number is divisible by 4

2. Practice case B (Hypotheses and Conclusions): In "If a figure is a square, then the figure is a rectangle," what is the conclusion?

Choices: the figure is a rectangle · a figure is a square · if · only if

Show solution
  1. The conclusion is the then-part.
  2. It is what follows if the hypothesis holds.
  3. Here the conclusion is the figure is a rectangle.

Answer: the figure is a rectangle

3. Practice case C (Hypotheses and Conclusions): What is the converse of "If a student scores at least 70, then the quiz is passed"?

Choices: If the quiz is passed, then a student scores at least 70. · If a student does not score at least 70, then the quiz is not passed. · If the quiz is not passed, then a student does not score at least 70. · a student scores at least 70 and the quiz is passed.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Hypotheses and Conclusions): What is the converse of "If a student scores at least 70, then the quiz is passed"?
  2. For a converse, switch the hypothesis and conclusion without negating them.
  3. The converse switches the hypothesis and conclusion.
  4. It does not negate them.
  5. So q -> p is the converse.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: If the quiz is passed, then a student scores at least 70.

4. Practice case D (Hypotheses and Conclusions): What is the contrapositive of "If a triangle is equilateral, then the triangle is isosceles"?

Choices: If the triangle is not isosceles, then a triangle is not equilateral. · If the triangle is isosceles, then a triangle is equilateral. · If a triangle is not equilateral, then the triangle is not isosceles. · If a triangle is equilateral, then the triangle is not isosceles.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Hypotheses and Conclusions): What is the contrapositive of "If a triangle is equilateral, then the triangle is isosceles"?
  2. For a contrapositive, switch the hypothesis and conclusion, then negate both parts.
  3. The contrapositive switches and negates both parts.
  4. p -> q becomes ¬q -> ¬p.
  5. That is the first choice.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: If the triangle is not isosceles, then a triangle is not equilateral.

5. Practice case E (Hypotheses and Conclusions): Lesson focus: Which transfer task would show readiness for Hypotheses and Conclusions?

Choices: Apply identify the if-part and then-part of a conditional in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.

Show solution
  1. A transfer task keeps the same objective but changes the surface context.
  2. Apply identify the if-part and then-part of a conditional in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply identify the if-part and then-part of a conditional in a new sentence or context.

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