The Lumenary · Grades 3–5

True, false, and, or, not

Name: ______________________

Date: ______________________

You will learn to

Decide whether a sentence is a statement, and whether a simple mathematical statement is true or false, giving a reason
Combine statements with "and", "or" and "not", reading "or" inclusively ("one or the other or both")
Use a single counterexample to show that an "every…" claim is false, and one example to show that a "some…" claim is true
Explain why checking many examples cannot prove a claim about every whole number
Say the opposite of "all are…" correctly as "at least one is not…"

1.
Which of these sentences are statements (sentences that are either true or false)? Choose all that are.
- A.5 + 5 = 10.
- B.5 + 5 = 11.
- C.Is 12 an even number?
- D.Draw a triangle.
- E.Every triangle has three sides.
2.
Which of these statements are true? Choose all that are.
- A.15 is odd and 15 is less than 10.
- B.15 is odd or 15 is less than 10.
- C.6 is even or 6 is a multiple of 3.
- D.8 is not bigger than 5.
3.
How many of the whole numbers from 1 to 20 make this statement true?

" $n$ is even or $n$ is a multiple of 5."
4.
How many of the whole numbers from 1 to 20 make this statement true?

" $n$ is even and $n$ is a multiple of 3."
5.
Kofi claims: "Every odd number from 11 to 25 is prime." Find all the counterexamples to his claim. Write them separated by commas, like {1, 2}.

(A prime number has exactly two factors: 1 and itself.)
6.
Zara checks the claim " $n \times n + n + 11$ is a prime number" for $n = 0, 1, 2, \dots, 9$ and gets 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 — all prime. What can she conclude?
- A.The claim is true for every whole number $n$ .
- B.The claim is true for $n = 0$ to $9$ , but she does not yet know about other numbers.
- C.The claim must be false, because patterns never last.
7.
Which sentence means the opposite of "All the pencils in the box are red"?
- A.None of the pencils in the box is red.
- B.At least one pencil in the box is not red.
- C.All the pencils in the box are blue.
- D.Some of the pencils in the box are red.
8.
The sign on the café door is wrong. It says: "We are open and we have mandazi today." Which sentence must be true?
- A.The café is closed and it has no mandazi.
- B.The café is closed, or it has no mandazi (or both).
- C.The café is closed.
9.
In Lean, ∀ n : Nat means "for every whole number $n$ ", → means "then" and ¬ means "not". Which Lean statement says: "Not every number that ends in 2 is divisible by 4"?
- A.¬ ∀ n : Nat, n % 10 = 2 → n % 4 = 0
- B.∀ n : Nat, n % 10 = 2 → ¬ (n % 4 = 0)
- C.22 % 10 = 2

Answer key — True, false, and, or, not

1.
The statements are 5 + 5 = 10 (true), 5 + 5 = 11 (false) and Every triangle has three sides (true). "Is 12 an even number?" is a question and "Draw a triangle" is an instruction: neither is true or false.
2.
- "15 is odd and 15 is less than 10": the second part is false, so the "and" statement is false.
- "15 is odd or 15 is less than 10": the first part is true, so it is true.
- "6 is even or 6 is a multiple of 3": both parts are true, so it is true ("or" is inclusive).
- "8 is not bigger than 5": 8 is bigger than 5, so this is false.
3.
The even numbers 2, 4, …, 20 give 10 numbers. The multiples of 5 are 5, 10, 15, 20; 10 and 20 are already on the list, so only 5 and 15 are new. That makes $10 + 2 = 12$ . Because "or" is inclusive, 10 and 20 (which make both parts true) count — but only once each.
4.
The multiples of 3 up to 20 are 3, 6, 9, 12, 15, 18. The even ones are 6, 12 and 18, so the answer is 3. (These are exactly the multiples of 6.)
5.
The odd numbers from 11 to 25 are 11, 13, 15, 17, 19, 21, 23, 25. Of these, $15 = 3 \times 5$ , $21 = 3 \times 7$ and $25 = 5 \times 5$ are not prime. So the counterexamples are 15, 21 and 25. Any one of them is enough to show that Kofi's claim is false.
6.
She has checked ten cases, so she knows the claim holds for $n = 0, \dots, 9$ — nothing more. A claim about every whole number needs a proof, or one counterexample to disprove it. Here $n = 10$ gives $10 \times 10 + 10 + 11 = 121 = 11 \times 11$ , which is not prime, so the claim is false.
7.
"All the pencils are red" is false as soon as one pencil is not red. So its opposite is "At least one pencil in the box is not red". "None is red" is too strong: in a box with two red pencils and one blue one, "all are red" and "none is red" are both false.
8.
An "and" statement is false when at least one of its parts is false. So the sign is wrong exactly when the café is closed, or it has no mandazi, or both. That is De Morgan's law: "not ( $P$ and $Q$ )" is the same as "(not $P$ ) or (not $Q$ )". We cannot say which part is false, so neither "closed" alone nor "closed and no mandazi" must be true.
9.
The answer is ¬ ∀ n : Nat, n % 10 = 2 → n % 4 = 0: "it is not true that every number ending in 2 is divisible by 4". Putting the ¬ inside instead, ∀ n, n % 10 = 2 → ¬ (n % 4 = 0), says that no number ending in 2 is divisible by 4 — which 12 shows is false. The difference is exactly the difference between "not all" and "none".