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If-Then Statements

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

A conditional says that whenever the hypothesis happens, the conclusion must follow. It does not say the hypothesis actually happened. Learning objective: Translate if-then claims as p → q. 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 (If-Then Statements): Worked example: Which rule or habit best matches If-Then Statements?

  1. If-Then Statements targets a specific reasoning habit.
  2. If-Then Statements focuses on translate if-then claims as p → q.
  3. The other choices either overclaim or change the logical relationship.

Answer: If-Then Statements focuses on translate if-then claims as p → q.

Practice problems

1. Practice case A (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): 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 (If-Then Statements): Lesson focus: Which transfer task would show readiness for If-Then Statements?

Choices: Apply translate if-then claims as p → q 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 translate if-then claims as p → q in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply translate if-then claims as p → q in a new sentence or context.

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