True and False Statements

Show your work for each problem.

1.
Which of these is a TRUE statement?
1. a) $5 + 3 = 8$
2. b) $5 + 3 = 6$
3. c) $5 + 3 = 7$
4. d) $5 + 3 = 9$
Answer: $5 + 3 = 8$
$5 + 3 = 8$ is true because 5 plus 3 equals 8. The other statements show wrong sums, so they are false.
2.
Which of these is NOT a statement?
1. a)A triangle has 3 sides
2. b) $7 + 2 = 9$
3. c) $7 + 2 = 10$
4. d)What is 7 + 2?
Answer: What is 7 + 2?
"What is 7 + 2?" is a question, not a statement. A question asks for information and cannot be true or false. The other three are statements: $7 + 2 = 9$ and "A triangle has 3 sides" are true, $7 + 2 = 10$ is false.
3.
Which number goes in the box to make the statement " $12 - \square = 7$ " true?
Answer: 5
$12 - 5 = 7$ , so the statement is true when the box holds 5. Any other number makes it false, for example $12 - 4 = 8$ .
4.
The statement " $4 \times 6 = 25$ " is false. Which number should replace 25 to make it true?
Answer: 24
$4 \times 6 = 24$ , not 25. So " $4 \times 6 = 24$ " is the true statement.
5.
Is the statement " $9 + 6 = 15$ " true or false?
Answer: True
1. What does the left side equal? Calculate $9 + 6$ . 15
2. What number is on the right side? 15
3. Are both sides equal? yes
4. So is the statement true or false? true
6.
Is the statement " $8 > 12$ " true or false?
Answer: False
1. What does the symbol $>$ mean? greater than
2. Is 8 greater than 12? no
3. So is the statement true or false? false
7.
Which sentence is a statement?
1. a)How many sides does a square have?
2. b)Count to ten!
3. c)Wow, that's amazing!
4. d)A hexagon has 6 sides.
Answer: A hexagon has 6 sides.
"A hexagon has 6 sides" is a statement: it makes a clear claim, and the claim is true. The command ("Count to ten!"), the question ("How many sides...") and the exclamation ("Wow...") cannot be true or false, so they are not statements.
8.
Which of these statements is FALSE?
1. a)A circle has no corners.
2. b) $10 \div 2 = 5$
3. c)All rectangles are squares.
4. d)All squares are rectangles.
Answer: All rectangles are squares.
"All rectangles are squares" is FALSE. A rectangle can have sides of different lengths (like 3 cm by 5 cm), so not every rectangle is a square. But every square IS a rectangle, because it has 4 right angles.
9.
Is the statement " $20 \leq 20$ " true or false?
1. a)True, because 20 is equal to 20
2. b)False, because 20 is not less than 20
3. c)False, because both sides must be different numbers
4. d)True, because 20 is less than 20
Answer: True, because 20 is equal to 20
$20 \leq 20$ is TRUE. The symbol $\leq$ means "less than or equal to". 20 is not less than 20, but it is equal to 20, and that is enough.
10.
Is "Close the door." a statement?
1. a)No, it is a question
2. b)Yes, it is a true statement
3. c)Yes, it is a false statement
4. d)No, it is a command, so it cannot be true or false
Answer: No, it is a command, so it cannot be true or false
"Close the door." is NOT a statement. It is a command: it tells someone to do something, and it cannot be true or false.
11.
Is the statement " $3 \times 7 = 20$ " true or false?
Answer: False
1. Calculate $3 \times 7$ . What is the result? 21
2. The statement says $3 \times 7 = 20$ . Is 21 equal to 20? no
3. So is the statement true or false? false
12.
Which statement is TRUE?
1. a)All prime numbers are odd.
2. b)Every number divisible by 2 is divisible by 4.
3. c)Every even number is divisible by 2.
4. d)Zero is a positive number.
Answer: Every even number is divisible by 2.
"Every even number is divisible by 2" is TRUE: that is exactly what even means. The others are false, and one counterexample each proves it: 6 is divisible by 2 but not by 4; 2 is a prime number but it is even; zero is neither positive nor negative.
13.
Is the statement "The sum of two odd numbers is always even" true or false?
1. a)True: the sum of two odd numbers is always even
2. b)Only sometimes true: it depends on the numbers
3. c)False: the sum of two odd numbers is always odd
4. d)Nobody can know without trying every pair of odd numbers
Answer: True: the sum of two odd numbers is always even
TRUE. Examples: $1 + 3 = 4$ , $5 + 7 = 12$ , $9 + 11 = 20$ . Examples alone do not prove "always", but this reason does: every odd number is an even number plus 1. Adding two odd numbers gives even + even + 1 + 1, and the two leftover 1s make 2, so the sum is always even.
14.
Is the statement "If a number ends in 0, it is divisible by 5" true or false?
Answer: True
1. $370 = 37 \times \square$ . What number goes in the box? 10
2. $10 = 2 \times \square$ . What number goes in the box? 5
3. Test one number: what is $370 \div 5$ ? 74
4. Every number ending in 0 is a multiple of 10, and 10 is a multiple of 5. Is the statement true or false? true
15.
Is the statement "All multiples of 6 are also multiples of 3" true or false?
Answer: True
1. $6 = 3 \times \square$ . What number goes in the box? 2
2. 24 is a multiple of 6. Write it as $24 = 3 \times \square$ : what goes in the box? 8
3. $42 = 6 \times 7$ is a multiple of 6. What is $42 \div 3$ ? 14
4. Every multiple of 6 is (a number) $\times 6$ = (a number) $\times 2 \times 3$ . Is the statement true or false? true