Biconditionals in Math
A free Logic lesson from the “Biconditionals and Definitions” unit, with a worked example and practice problems including step-by-step solutions.
An iff definition requires two directions: every named object has the defining property, and every object with the property belongs to the named class. Test a proposed definition with examples and near-miss non-examples. A four-equal-side condition, for example, does not distinguish squares from other rhombi.
What you'll learn
- Find biconditional structure in even numbers, equality, and geometry definitions
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Over the integers, n is even iff n=2k for some integer k. What must a proof of this biconditional establish?
- Iff requires each direction.
- One example cannot establish a universal definition or equivalence.
Answer: Both implications
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Over the integers, n is even iff n=2k for some integer k. What must a proof of this biconditional establish?
Choices: Both implications · Only even→2k · Only 2k→even · One numerical example
Show solution
- Iff requires each direction.
- One example cannot establish a universal definition or equivalence.
Answer: Both implications
2. Is “a square is a figure with four equal sides” a sufficient definition among quadrilaterals?
Choices: Yes, equal sides force right angles · No, squares have unequal sides · Yes, every quadrilateral is a square · No; a non-square rhombus is a counterexample
Show solution
- Equal sides do not force right angles.
- The proposed definition admits objects outside the intended class.
Answer: No; a non-square rhombus is a counterexample
3. Using the inclusive definition, which description defines a rectangle?
Choices: A quadrilateral with one right angle · A four-sided shape that is not a square · A quadrilateral with four right angles · A quadrilateral with four equal sides
Show solution
- Four right angles specify rectangles and include squares.
- Equal sides alone describe rhombi, not exactly rectangles.
Answer: A quadrilateral with four right angles
4. For integers n, “n is divisible by 25 iff n is even” is true for every n.
Choices: True · False
Show solution
- The integer 2 is even but is not divisible by 25.
- Failure of one direction disproves the biconditional.
Answer: False
5. Which formula translates “p if and only if q”?
Choices: (p → q) ∧ (q → p) · (p → q) ∨ (q → p) · p ∧ ¬q · p ∨ q
Show solution
- Both directions must hold, so join them with and.
Answer: (p → q) ∧ (q → p)
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