If-Then Statements

A **conditional statement**, also called an **if-then statement**, joins two statements with the words **if** and **then**: $$\text{If } p\text{, then } q.$$ - The part after "if" is the **hypothesis** ($p$), the condition. - The part after "then" is the **conclusion** ($q$), what must follow. In symbols we write $p \to q$ and read it as "if $p$, then $q$" or "$p$ implies $q$". **Example:** If a number ends in 0, then it is divisible by 10. - Hypothesis $p$: a number ends in 0 - Conclusion $q$: the number is divisible by 10 Think of a conditional as a **promise**. The promise is broken only when the hypothesis happens and the conclusion does not. So $p \to q$ is **false in exactly one case**: $p$ is true and $q$ is false. In all other cases it is true. | $p$ | $q$ | $p \to q$ | |:---:|:---:|:---:| | T | T | T | | T | F | F | | F | T | T | | F | F | T | The interactive truth table in this lesson lists the same four rows in the opposite order, starting with F, F. Both orders are fine. An example where the hypothesis is true and the conclusion is false is called a **counterexample**. One counterexample is enough to show that a conditional is false.

Finding the Hypothesis and the Conclusion

Identify the hypothesis and the conclusion: "If a triangle has a right angle, then it is a right triangle."

Find the part after "if": "a triangle has a right angle" = Hypothesis $p$: a triangle has a right angle

Find the part after "then": "it is a right triangle" = Conclusion $q$: the triangle is a right triangle

Write it in symbols: The condition comes first, the result second = $p \to q$

Answer: Hypothesis: a triangle has a right angle. Conclusion: it is a right triangle. In symbols: $p \to q$.

Rewriting an Everyday Sentence

Rewrite in if-then form: "I take an umbrella whenever it rains."

Find the condition: The word "whenever" introduces the condition: it rains = Hypothesis: it rains

Find what follows from the condition: What happens when it rains? I take an umbrella = Conclusion: I take an umbrella

Put the hypothesis after "if" and the conclusion after "then": The sentence started with the conclusion, so we change the order = If it rains, then I take an umbrella.

Answer: If it rains, then I take an umbrella. The hypothesis does not always come first in the original sentence.

Rewriting a Math Sentence and Checking It

Rewrite "Every multiple of 10 is even" in if-then form and check it with the number 70.

Find what we start from: We start from a number that is a multiple of 10 = $p$: a number is a multiple of 10

Find what is claimed about it: Such a number is even = $q$: the number is even

Write the conditional: If $p$, then $q$ = If a number is a multiple of 10, then it is even.

Check with 70: $70 = 10 \times 7$, so the hypothesis is true. $70 \div 2 = 35$, so 70 is even = The conclusion is true for 70

Answer: If a number is a multiple of 10, then it is even. For 70 both parts are true, so this case agrees with the statement.

When Is a Promise Broken?

A teacher says: "If you score at least 80 points, then you get a sticker." Four students: Ana scores 85 and gets a sticker, Ben scores 90 and gets no sticker, Cleo scores 70 and gets a sticker, Dan scores 60 and gets no sticker. Who shows that the teacher broke the promise?

Mark the hypothesis $p$ for each student: $p$: score $\geq 80$. Ana $85$, Ben $90$: true. Cleo $70$, Dan $60$: false = $p$ is true only for Ana and Ben

Mark the conclusion $q$ for each student: $q$: gets a sticker. Ana and Cleo: true. Ben and Dan: false = $q$ is false for Ben and Dan

Look for the only false case: $p$ true and $q$ false: Ana: T, T. Ben: T, F. Cleo: F, T. Dan: F, F = Only Ben has T and F

Answer: Only Ben breaks the promise: he scored at least 80 but got no sticker. Cleo getting a sticker with 70 points does not break it, because the promise says nothing about scores below 80.

Disproving a Statement with a Counterexample

Is this statement true? "If $x^2 = 25$, then $x = 5$."

Look for a value that makes the hypothesis true: Which numbers squared give 25? $5^2 = 25$ and $(-5)^2 = 25$ = $x = 5$ or $x = -5$

Test the conclusion for each value: For $x = 5$ the conclusion $x = 5$ is true. For $x = -5$ the conclusion $x = 5$ is false = $x = -5$ gives T and F

Decide: One case with a true hypothesis and a false conclusion is enough = $x = -5$ is a counterexample

Answer: The statement is false. The counterexample $x = -5$ makes the hypothesis $x^2 = 25$ true but the conclusion $x = 5$ false.

If-then statements are everywhere, often hidden in ordinary sentences: - **Shopping**: "If your order is at least 40 euros, then delivery is free." - **Rules and games**: "If you roll a 6, then you roll again." - **Science**: "If water is heated to 100 °C at sea level, then it boils." - **Programming**: every app is full of commands like "if the password is correct, then open the account". - **Mathematics**: almost every rule and theorem is a conditional, for example "if a triangle has a right angle, then $a^2 + b^2 = c^2$" (where $c$ is the longest side). In the lesson Deductive Reasoning you used if-then rules as premises. Now you will learn how they are built, how to write them yourself and how to test them. This is the starting point for the converse, the inverse and the contrapositive, which come next in the lessons Converse and Inverse and Contrapositive.

Mistake: Taking the first part of the sentence as the hypothesis, as in "I take an umbrella if it rains" → hypothesis "I take an umbrella"

Why: The hypothesis is the part that follows the word "if" (or "when", "whenever"), wherever it stands in the sentence.

Correct: Find the condition first: "it rains". Then write: If it rains, then I take an umbrella.

Mistake: Thinking that $p \to q$ is false whenever the hypothesis is false

Why: A conditional only promises something for the case where $p$ happens. If $p$ does not happen, the promise cannot be broken.

Correct: $p \to q$ is false only in the case T → F. When $p$ is false, the conditional is true, whatever $q$ is.

Mistake: Using a number that does not satisfy the hypothesis as a counterexample, for example $x = 0$ against "if $x > 3$, then $x > 1$"

Why: For $x = 0$ the conclusion $x > 1$ is false, but the hypothesis $x > 3$ is false too, so this case cannot break the statement.

Correct: A counterexample must make the hypothesis true and the conclusion false. For "if $x > 3$, then $x > 1$" no such number exists, so the statement is true.

Mistake: Believing that one example that works proves the statement

Why: "If $n$ is a whole number, then $2^n + 1$ is prime" works for $n = 1$ ($3$) and $n = 2$ ($5$), but $n = 3$ gives $9$, which is not prime.

Correct: Examples can only support a statement. To prove it you need a general argument, but a single counterexample is enough to disprove it.

Online Shopping

Shops use if-then rules for free delivery, discounts and gifts. To use the offer, you must make the hypothesis true.

"If your order is at least 40 dollars, then delivery is free." An order of 52 dollars makes the hypothesis true, so delivery must be free.

Testing Claims in Mathematics

Mathematicians test a conditional by searching for a counterexample. If they find one, the claim is false.

"If a number is prime, then it is odd" is false, because 2 is prime and even.

Smart Homes and Programming

A thermostat runs on a conditional: if the room is colder than a set value, then the heating turns on.

"If the temperature is below 18 °C, then the heating turns on." At 17 °C the hypothesis is true, so the heating must turn on. At 20 °C nothing is promised.

A conditional statement has the form "if $p$, then $q$", written $p \to q$

The hypothesis $p$ is the condition after "if"; the conclusion $q$ is the result after "then"

Many sentences hide a conditional: "every A is B" means "if something is A, then it is B"

$p \to q$ is false only when $p$ is true and $q$ is false (T → F); in the other three cases it is true

A counterexample makes the hypothesis true and the conclusion false, and one is enough to disprove a statement

Examples that work support a statement, but they do not prove it

Q: Why is "if $p$, then $q$" true when $p$ is false?

A: Because the statement only makes a promise about the case where $p$ happens. If you say "if it rains, I take an umbrella" and it does not rain, you have not broken your word, whatever you do.

Q: Does the word "then" always have to be there?

A: No. "If it rains, I stay home", "I stay home when it rains" and "Rain means I stay home" are all the same conditional. Writing it with "if" and "then" just makes the hypothesis and the conclusion easy to see.

Q: How many counterexamples do I need to show a statement is false?

A: Just one. A conditional claims that the conclusion follows every time the hypothesis is true, so a single case with T → F is enough to break it.

Q: What happens if I swap the hypothesis and the conclusion?

A: You get a new statement that can have a different truth value. "If a shape is a square, then it is a rectangle" is true, but "if a shape is a rectangle, then it is a square" is false. Swapping and negating parts is the topic of the next lessons, Converse and Inverse and Contrapositive.

Exit Presentation

If-Then Statements

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Logic & Sets / Logical Reasoning

If-Then Statements

Learn to write conditional statements in if-then form, find the hypothesis and conclusion, and decide when a conditional is false.

Intermediate25 min

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