Mathwords logoMathwords

Conditional

Conditional

An "If . . . then . . ." statement. For example, "If it is raining, then the grass is wet."

 

 

See also

Hypothesis, conclusion, converse, inverse, contrapositive, biconditional

Key Formula

p→qp \rightarrow q
Where:
  • pp = The hypothesis (the condition that comes after "if")
  • qq = The conclusion (the result that comes after "then")
  • →\rightarrow = The conditional arrow, read as "implies" or "if . . . then"

Example

Problem: Given the conditional statement "If a number is divisible by 6, then it is divisible by 3," identify the hypothesis and conclusion, then state the converse, inverse, and contrapositive.
Identify the parts: The hypothesis p is "a number is divisible by 6." The conclusion q is "it is divisible by 3."
p→qp \rightarrow q
Write the converse: Swap the hypothesis and conclusion: "If a number is divisible by 3, then it is divisible by 6."
q→pq \rightarrow p
Write the inverse: Negate both parts of the original: "If a number is NOT divisible by 6, then it is NOT divisible by 3."
¬p→¬q\lnot p \rightarrow \lnot q
Write the contrapositive: Swap and negate both parts: "If a number is NOT divisible by 3, then it is NOT divisible by 6."
¬q→¬p\lnot q \rightarrow \lnot p
Check truth values: The original conditional is true (every multiple of 6 is a multiple of 3). The contrapositive is also true — it is always logically equivalent to the original. However, the converse is false: 9 is divisible by 3 but not by 6. The inverse is also false for the same reason.
Answer: Hypothesis: "a number is divisible by 6." Conclusion: "it is divisible by 3." The contrapositive is true, but the converse and inverse are both false.

Another Example

Problem: Determine when the conditional "If x > 10, then x > 5" is false.
Recall the truth table rule: A conditional p → q is false in exactly one case: when p is true and q is false.
Test p true, q false: You would need a number greater than 10 that is NOT greater than 5. No such number exists.
Conclude: Since you can never have p true and q false at the same time, the conditional is always true.
Answer: The conditional is never false — it is true for all values of x.

Frequently Asked Questions

When is a conditional statement false?
A conditional p → q is false only when the hypothesis p is true and the conclusion q is false. In every other case — including when the hypothesis itself is false — the conditional is considered true. This surprises many students, but it follows directly from the logical definition: the statement only makes a promise about what happens when p is true.
What is the difference between a conditional and a biconditional?
A conditional p → q is a one-way "if . . . then" statement. A biconditional p ↔ q means "p if and only if q," which requires the conditional to work in both directions: p → q AND q → p must both be true. A biconditional is true only when p and q have the same truth value.

Conditional vs. Contrapositive

The conditional states "If p, then q" (p → q). The contrapositive states "If not q, then not p" (¬q → ¬p). These two statements are logically equivalent — they always have the same truth value. By contrast, the converse (q → p) and inverse (¬p → ¬q) are equivalent to each other but NOT necessarily equivalent to the original conditional.

Why It Matters

Conditional statements are the backbone of mathematical reasoning and proof. Nearly every theorem is expressed as a conditional: "If these conditions hold, then this result follows." Understanding when a conditional is true or false, and how it relates to its converse, inverse, and contrapositive, is essential for constructing valid arguments in geometry, algebra, and beyond.

Common Mistakes

Mistake: Assuming the converse is automatically true whenever the original conditional is true.
Correction: The converse swaps the hypothesis and conclusion, which can change the truth value. "If it is raining, then the ground is wet" does not mean "If the ground is wet, then it is raining" — a sprinkler could be the cause. Always verify the converse separately.
Mistake: Thinking a conditional is false when the hypothesis is false.
Correction: A conditional with a false hypothesis is always true (this is called "vacuously true"). The conditional only fails when the hypothesis is true and the conclusion is false.

Related Terms