Question 1
Suppose , and . Use proof by contradiction to show .
- Given: and .
- Goal: Show that using proof by contradiction.
- Method: In a proof by contradiction, we assume the opposite (negation) of the desired result () and show that it leads to a logical impossibility or contradicts a given condition.
Step 1 · Assume the Negation of the Statement
To prove by contradiction, assume the opposite is true:
Step 2 · Combine Inequalities to Reach a Contradiction
We are given:
Adding the assumption to :
This contradicts the given statement that:
Since our assumption leads to a contradiction, the assumption must be false.
Therefore:
Hence, by proof by contradiction, .
- Incomplete Negation: Incorrectly assuming the opposite of is only , forgetting the equality case (the complete negation is ).
- Missing Contradiction Link: Not explicitly comparing the derived inequality () with the given equation () to state where the contradiction occurs.
More questions in A1.6
Suppose , and . Use proof by contradiction to show .
Let be a rational number and be an irrational number. Use proof by contradiction to show that is an irrational number.
Use proof by contradiction to prove that if for an integer , is even, then so is .
[Hint : Assume is not even, that is, it is of the form , for some integer , and then proceed.]
Use proof by contradiction to prove that if for an integer , is divisible by 3, then is divisible by 3.
Use proof by contradiction to show that there is no value of for which ends with the digit zero.
Prove by contradiction that two distinct lines in a plane cannot intersect in more than one point.