Mathematics
Grade 10
15 min
Counter-examples
Counter-examples
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Define the term 'counter-example' and explain its role in mathematical logic.
Distinguish between a supporting example and a counter-example.
Analyze a universal mathematical statement to determine if it is true or false.
Find a valid counter-example to disprove a false universal statement in algebra.
Construct a valid counter-example to disprove a false universal statement in geometry.
Articulate why a single counter-example is sufficient to disprove a universal conjecture.
Is it true that all four-sided shapes with four equal sides are squares? 🤔 Let's find out!
In this tutorial, you will learn about one of the most powerful tools in mathematics: the counter-example. A counter-example is a single instance that proves a general statement...
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Key Concepts & Vocabulary
TermDefinitionExample
ConjectureA mathematical statement that is believed to be true but has not yet been formally proven.Goldbach's Conjecture states that every even integer greater than 2 is the sum of two prime numbers (e.g., 10 = 3 + 7). This has not been proven true for all such integers.
Universal StatementA statement that claims a property is true for all elements in a set. It often uses words like 'all', 'every', or 'for any'.'All prime numbers are odd.' This claims a property (being odd) is true for the entire set of prime numbers.
Counter-exampleA specific example that shows a universal statement or conjecture is false.For the statement 'All prime numbers are odd,' the number 2 is a counter-example because 2 is a prime numbe...
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Core Formulas
The Principle of Disproof
To disprove the statement 'For all x, P(x) is true', you must find a single value c such that P(c) is false.
This is the fundamental rule for using counter-examples. You don't need to show it's false for many values; one is enough to break a universal rule.
Validity of a Counter-example
To disprove 'If H, then C', the counter-example must satisfy H but not satisfy C.
A valid counter-example must meet the initial conditions (the hypothesis, H) but fail to meet the outcome (the conclusion, C).
4 more steps in this tutorial
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Challenging
Consider the conjecture: 'For any positive integer n, 2ⁿ > n²'. Which of the following is a valid counter-example?
A.n = 1
B.n = 5
C.n = 3
D.n = 0
Challenging
Which of the following conjectures is FALSE?
A.For any two distinct points, there is exactly one line that contains them.
B.If a quadrilateral is a rhombus, its diagonals are perpendicular.
C.The sum of the squares of two sides of a triangle is equal to the square of the third side.
D.For any integer n > 1, there is at least one prime number p such that n < p < 2n.
Challenging
A student is testing the conjecture: 'If a quadrilateral has a pair of equal opposite angles and a pair of equal opposite sides, then it must be a parallelogram.' Which figure would be a valid counter-example?
A.rhombus
B.An isosceles trapezoid
C.kite
D.rectangle
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