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💡 What Is a Counterexample?
A counterexample is a single example that proves a claim is false.
If someone says "ALL swans are white," you don't need to check every swan in the world. You just need to find one black swan — and the claim is disproved!
In math, it works the same way. If someone claims something is ALWAYS true, finding just one case where it fails is enough to prove them wrong.
Claim: "All prime numbers are odd."
🧪 Let's check some primes:
3 is odd ✓ 5 is odd ✓ 7 is odd ✓ 11 is odd ✓ 13 is odd ✓
🔍 But wait…
2 is a prime number, and 2 is even!
❌ Counterexample: 2 — The claim is FALSE!
Claim: "The square of any number is bigger than that number."
🧪 Let's check:
3² = 9 > 3 ✓ 5² = 25 > 5 ✓ 10² = 100 > 10 ✓
🔍 But wait…
1² = 1, and 1 is NOT bigger than 1! Also 0² = 0.
❌ Counterexample: 1 (or 0) — The claim is FALSE!
1 Find the Counterexample! 🔎
Each claim below is false. Find a counterexample to disprove it.
a"All multiples of 3 are odd."
My counterexample: because
b"If you subtract two numbers, you always get a smaller number."
My counterexample: because
c"All numbers that end in 5 are divisible by 10."
My counterexample: because
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2 True or False? You Decide! ⚖️
For each claim: if you think it's true, explain why. If you think it's false, give a counterexample.
a"Every even number greater than 2 can be divided by 2 with no remainder."
My answer:
Reason / counterexample:
b"The product of two odd numbers is always even."
My answer:
Reason / counterexample:
c"If a number is divisible by 6, it is also divisible by 3."
My answer:
Reason / counterexample:
d"Adding 1 to a prime number always gives a composite number."
My answer:
Reason / counterexample:
3 Make Your Own False Claim! 🎭
Write a math claim that SOUNDS true but is actually false. Then give the counterexample.
My false claim:
Counterexample:
Answer Key:
1a) 6 is a multiple of 3 and is even — 1b) 5 − 5 = 0, not smaller (or 3 − 7 = −4) — 1c) 15 ends in 5 but 15 ÷ 10 = 1.5 —
2a) TRUE (that's the definition of even) — 2b) FALSE (3 × 5 = 15, which is odd) — 2c) TRUE (6 = 2 × 3, so any multiple of 6 is a multiple of 3) — 2d) FALSE (2 + 1 = 3, which is prime)