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Sufficient Conditions

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

A sufficient condition gives enough information to force the conclusion, even if there may be other ways to reach that conclusion. Learning objective: Recognize conditions that are enough to guarantee a result. 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 (Sufficient Conditions): In "If a number is divisible by 4, then the number is even," what is the hypothesis?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Sufficient Conditions): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The hypothesis is the if-part.
  4. The conclusion is the then-part.

Answer: a number is divisible by 4

Practice problems

1. Practice case A (Sufficient Conditions): 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 (Sufficient Conditions): In "If a number is divisible by 4, then the number is even," what is the hypothesis?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  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 satisfies the original prompt and that the other choices fail the same test.

Answer: a number is divisible by 4

2. Practice case B (Sufficient Conditions): 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 (Sufficient Conditions): 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 not a student scores at least 70, then not the quiz is passed. · If not the quiz is passed, then not a student scores 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 (Sufficient Conditions): What is the converse of "If a student scores at least 70, then the quiz is passed"?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  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 satisfies the original prompt and that the other choices fail the same test.

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

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

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Sufficient Conditions): What is the contrapositive of "If a triangle is equilateral, then the triangle is isosceles"?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  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 satisfies the original prompt and that the other choices fail the same test.

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

5. Practice case E (Sufficient Conditions): If p is False and q is True, what is p → q?

Choices: True · False

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Sufficient Conditions): If p is False and q is True, what is p → q?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A conditional is false only when p is true and q is false.
  4. Here p is False and q is True.
  5. So p → q is True.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: True

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