Conditional Statement
In short: A conditional statement ties two claims together in an if-then sentence, promising that whenever the first claim, the hypothesis, is true, the second, the conclusion, is true as well.
GeometryGeometryG-CO.C.9
What is Conditional Statement?
A conditional statement makes a promise: when the condition after if is met, the result after then must follow. In logic it reads if \(p\), then \(q\), with \(p\) as the hypothesis and \(q\) as the conclusion.
The promise is broken in just one situation, namely a true hypothesis paired with a false conclusion. A single case of that kind, called a counterexample, kills the sentence. Showing that it holds takes an argument that covers every possible case.
How to spot it
- A sentence contains if and then, or can be reworded that way, such as every square is a rectangle.
- A proof starts from a given fact and tries to reach a stated result, which is the hypothesis and conclusion pairing.
- The problem asks you to decide whether a claim is true or to find a counterexample.
- A diagram shows one shaded region sitting entirely inside another region.
The method
- Mark the hypothesis (the clause after if) with one underline and the conclusion (the clause after then) with two.
- When the claim is phrased loosely, reword it so the condition comes first and the result second.
- Hunt for a counterexample: an instance that meets the condition yet misses the result.
- If testing finds none, give a reason why the result must follow; if one turns up, call the conditional statement false.
Conditional Statement: solved example
Problem
Rewrite “Every whole number that ends in 4 is divisible by 4” as an if-then sentence, name its parts and decide whether it is true.
Solution
If-then form: If a whole number ends in 4, then it is divisible by 4. The hypothesis is “a whole number ends in 4”; the conclusion is “it is divisible by 4”.
Test cases: \(24 \div 4 = 6\) works, but \(14 \div 4 = 3.5\) does not. The number 14 makes the hypothesis true and the conclusion false, so it is a counterexample and the statement is false.
Try it
Question 1
Name the two parts of: “If a polygon has 6 sides, then it is a hexagon.”
Show the answer
Hypothesis: the polygon has 6 sides. Conclusion: it is a hexagon.
Question 2
Decide and justify: “If a whole number is a multiple of 8, then 4 divides it.”
Show the answer
True. Writing the number as \(8k = 4(2k)\) shows 4 always divides it, whatever \(k\) is.
Conditional Statement vs. Converse
Reading left to right, the original runs from hypothesis to conclusion. The converse flips that direction. Squares-are-rectangles holds, yet rectangles-are-squares fails, because a 3 by 5 rectangle has unequal sides. A true conditional statement does not guarantee a true converse.
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