If-Then Statements

Show your work for each problem.

1.
What is the hypothesis of the statement "If it rains, then the football match is cancelled"?
1. a)It rains
2. b)The football match is cancelled
3. c)The match is not cancelled
4. d)It does not rain
Answer: It rains
The hypothesis is the condition after "if": it rains. "The football match is cancelled" is the conclusion.
2.
Complete the conditional so that it is true: if $x = 4$ , then $3x - 2 = \square$ . Which number goes in the box?
Answer: 10
$3 \times 4 - 2 = 12 - 2 = 10$ , so the true conditional is: if $x = 4$ , then $3x - 2 = 10$ .
3.
Check the conditional "If $x = 6$ , then $2x + 3 = 15$ " by computing $2x + 3$ for $x = 6$ .
Answer: 15
1. What is $2 \times 6$ ? 12
2. What is $12 + 3$ ? 15
4.
Which sentence is "All squares are rectangles" written in if-then form?
1. a)If a shape has four sides, then it is a square
2. b)If a shape is a rectangle, then it is a square
3. c)If a shape is a square, then it is a rectangle
4. d)If a shape is not a square, then it is not a rectangle
Answer: If a shape is a square, then it is a rectangle
The sentence starts from squares, so the hypothesis is "a shape is a square" and the conclusion is "it is a rectangle". The second choice swaps the two parts, which gives a different (and false) statement.
5.
The truth table of $p \to q$ has 4 rows. In how many of these rows is $p \to q$ false?
Answer: 1
Only the row with $p$ true and $q$ false gives a false conditional. The other 3 rows are true.
6.
A shop's rule: if your order is at least 30 dollars, then delivery is free. You buy a book for 12 dollars and a game for 15 dollars. How many dollars more must you spend so that delivery is free?
Answer: 3
1. What is your total in dollars so far? 27
2. How many dollars are missing to reach 30? 3
7.
In which case is the conditional $p \to q$ false?
1. a) $p$ false, $q$ true
2. b) $p$ true, $q$ false
3. c) $p$ true, $q$ true
4. d) $p$ false, $q$ false
Answer: $p$ true, $q$ false
A conditional is broken only when the hypothesis happens and the conclusion does not: $p$ true and $q$ false. When $p$ is false, the conditional is true whatever $q$ is.
8.
The statement "If a whole number is greater than 5, then it is greater than 8" is false. What is the smallest whole number that is a counterexample?
Answer: 6
A counterexample makes the hypothesis true ( $n > 5$ ) and the conclusion false ( $n \leq 8$ ). The numbers 6, 7 and 8 do this, and the smallest is 6.
9.
Claim: if a whole number is divisible by 2, then it is divisible by 4. Find the smallest positive counterexample.
Answer: 2
1. What is the smallest positive whole number that is divisible by 2? 2
2. What is the remainder when you divide 2 by 4? 2
3. So which number is the smallest counterexample? 2
10.
Mia says: "If I finish my homework, then I go to the cinema." She did not finish her homework, but she went to the cinema anyway. Is her statement true or false?
1. a)We cannot say, because the hypothesis is false
2. b)False, because she did not finish her homework
3. c)True, because the hypothesis is false, so the promise is not broken
4. d)False, because she went to the cinema
Answer: True, because the hypothesis is false, so the promise is not broken
The hypothesis "I finish my homework" is false and the conclusion "I go to the cinema" is true. That is the case F → T, and a conditional is true in that case. Mia only promised what happens when she finishes her homework.
11.
A shop's rule: if you buy at least 3 notebooks, then each notebook costs 2 dollars instead of 3 dollars. You buy 4 notebooks. How many dollars do you pay in total?
Answer: 8
$4 \geq 3$ , so the hypothesis is true and the conclusion applies: each notebook costs 2 dollars. $4 \times 2 = 8$ dollars.
12.
Statement: if a number is a multiple of 6, then it is a multiple of 3. Check the statement for the number 42: first divide 42 by 6, then by 3. What is $42 \div 3$ ?
Answer: 14
1. Check the hypothesis: what is $42 \div 6$ ? 7
2. The hypothesis is true. Now check the conclusion: what is $42 \div 3$ ? 14
13.
Rule: if the temperature is below 18 °C, then the heating turns on. At 7:00 the room is 23 °C, and the temperature drops by 2 °C every hour. After how many full hours does the heating turn on for the first time?
Answer: 3
1. What is the temperature after 1 hour, in °C? 21
2. What is the temperature after 2 hours, in °C? 19
3. What is the temperature after 3 hours, in °C? 17
4. After how many hours is the temperature below 18 °C for the first time? 3
14.
Which number is a counterexample to the statement "If a number is prime, then it is odd"?
1. a)3
2. b)2
3. c)15
4. d)9
Answer: 2
2 is prime, because its only divisors are 1 and 2, so the hypothesis is true. But 2 is even, so the conclusion is false. 3 is prime and odd, and 9 and 15 are not prime, so none of them breaks the statement.
15.
Claim: if $n$ is a whole number greater than 1, then $n^2 + n + 1$ is prime. What is the smallest counterexample $n$ ?
Answer: 4
$n = 2$ : $4 + 2 + 1 = 7$ , prime. $n = 3$ : $9 + 3 + 1 = 13$ , prime. $n = 4$ : $16 + 4 + 1 = 21 = 3 \times 7$ , not prime. So $n = 4$ makes the hypothesis true and the conclusion false.
16.
Claim: if $n$ is a positive whole number, then $2^n + 1$ is prime. Test $n = 1, 2, 3$ and find the smallest counterexample.
Answer: 3
1. What is $2^1 + 1$ ? 3
2. What is $2^2 + 1$ ? 5
3. What is $2^3 + 1$ ? 9
4. 3 and 5 are prime, but 9 is not. For which $n$ is the conclusion false? 3
17.
Statement: if $x > 2$ , then $x^2 > 9$ . Which of the whole numbers 3, 4 and 5 is a counterexample?
Answer: 3
1. All three numbers are greater than 2, so the hypothesis is true for each. What is $3^2$ ? 9
2. What is $4^2$ ? 16
3. What is $5^2$ ? 25
4. Which of 3, 4 and 5 gives a square that is NOT greater than 9? 3
18.
Four cards each have a number on one side and a color on the other. You see: 4, 7, red, blue. Rule: if a card has an even number on one side, then its other side is red. Which cards must you turn over to check the rule, and how many are they?
Answer: 2
1. Only an even number makes the hypothesis true. Which number card must you turn over? Enter its number. 4
2. A blue card breaks the rule if it hides an even number. Counting the 4 and the blue card, how many cards must you turn over? 2