Equivalent statements
Understand equivalent statements for Year 11 Specialist Mathematics in Queensland (QCAA). This sub-topic introduces the language of proof: the implication "if P then Q", its converse, and the equivalence "P if and only if Q".
You will learn to write a converse correctly, test both directions of a biconditional, and use the symbols for implication, equivalence and equality — the logical foundation for every proof later in the course.
Theory
Equivalent statements are the heart of proof in Year 11 Specialist Mathematics (QCAA, Queensland). This page explains the implication \(P\Rightarrow Q\), its converse \(Q\Rightarrow P\), and the equivalence (biconditional) \(P\Leftrightarrow Q\), which holds only when both directions are true — plus the symbols \(\Rightarrow\), \(\Leftrightarrow\) and \(=\).
An implication \(P\Rightarrow Q\) reads "if \(P\) then \(Q\)": it claims that whenever the hypothesis \(P\) is true, the conclusion \(Q\) must be true as well. Here \(P\) is the hypothesis and \(Q\) is the conclusion.
The converse of \(P\Rightarrow Q\) is \(Q\Rightarrow P\). It simply swaps the hypothesis and the conclusion — nothing is negated. A true implication does not make its converse true: "if a number is a multiple of \(4\), then it is even" is true, but "if a number is even, then it is a multiple of \(4\)" is false.
An equivalence (or biconditional) \(P\Leftrightarrow Q\) reads "\(P\) if and only if \(Q\)". It is true only when both \(P\Rightarrow Q\) and \(Q\Rightarrow P\) are true. When that happens, \(P\) and \(Q\) are equivalent statements: they always have the same truth value.
The symbols matter: \(\Rightarrow\) means "implies" (one direction), \(\Leftrightarrow\) means "is equivalent to" (both directions), and \(=\) links values such as \(x=2\), not whole statements.
The biconditional is exactly the two one-way implications together:
An implication is logically the same as its contrapositive (negate and swap), so this is the standard "equivalent statement":
How to check whether two statements are equivalent
- Name the hypothesis \(P\) and the conclusion \(Q\), and write the claim as \(P\Rightarrow Q\).
- Test the forward direction \(P\Rightarrow Q\): does every case with \(P\) true also have \(Q\) true? If not, find a counterexample.
- Test the converse \(Q\Rightarrow P\) separately — swap the two parts, then look for a case where \(Q\) holds but \(P\) fails.
- Conclude: if both directions hold, write \(P\Leftrightarrow Q\) (equivalent); if either direction fails, they are not equivalent.
Name the parts, then swap them to form the converse:
| \(P\) | \(:\) | \(n \text{ is a multiple of } 4\) |
| \(Q\) | \(:\) | \(n \text{ is even}\) |
| \(\text{statement}\) | \(:\) | \(P \Rightarrow Q\) |
| \(\text{converse}\) | \(:\) | \(Q \Rightarrow P\) |
Test the converse with a value that is even:
| \(\text{try } n\) | \(=\) | \(6\) |
| \(6\) | \(\text{is}\) | \(\text{even, so } Q \text{ holds}\) |
| \(6 \div 4\) | \(=\) | \(1.5 \text{ (not a whole number)}\) |
| \(\text{so } P\) | \(\text{is}\) | \(\text{false here}\) |
Converse: "if \(n\) is even, then \(n\) is a multiple of \(4\)" — false, since \(6\) is a counterexample.
Forward direction — every multiple of \(4\) is even:
| \(\text{multiple of } 4 \Rightarrow \text{even}\) | \(:\) | \(\text{true}\) |
Converse direction — test an even number:
| \(\text{try } n\) | \(=\) | \(6\) |
| \(6\) | \(:\) | \(\text{even, but } 6 \div 4 = 1.5\) |
| \(\text{even} \Rightarrow \text{multiple of } 4\) | \(:\) | \(\text{false}\) |
One direction fails, so the biconditional cannot hold:
| \(P \Leftrightarrow Q\) | \(:\) | \(\text{false}\) |
No — the forward direction holds but the converse fails at \(6\), so it is not an equivalence.
An implication is equivalent to its contrapositive (negate and swap), not its converse:
| \(\text{statement}\) | \(:\) | \(P \Rightarrow Q\) |
| \(\text{contrapositive}\) | \(:\) | \(\lnot Q \Rightarrow \lnot P\) |
| \(:\) | \(\text{not div } 3 \Rightarrow \text{not div } 9\) |
Equivalent statement: "if a number is not divisible by \(3\), then it is not divisible by \(9\)".
Check which directions are true, then pick the matching symbol:
| \(n \text{ even} \Rightarrow n^2 \text{ even}\) | \(:\) | \(\text{true}\) |
| \(n^2 \text{ even} \Rightarrow n \text{ even}\) | \(:\) | \(\text{true}\) |
| \(\text{both directions hold}\) | \(\Rightarrow\) | \(\text{use } \Leftrightarrow\) |
| \(=\) | \(:\) | \(\text{links values, not statements}\) |
The correct symbol is \(\Leftrightarrow\): the two statements are equivalent.
Common pitfalls
Frequently asked questions
What is the difference between an implication and an equivalence?
An implication \(P\Rightarrow Q\) asserts one direction: if \(P\) then \(Q\). An equivalence \(P\Leftrightarrow Q\) asserts both directions at once, so it is true only when \(P\Rightarrow Q\) and \(Q\Rightarrow P\) both hold.
How do you find the converse of a statement?
Swap the hypothesis and the conclusion. The converse of "if \(P\) then \(Q\)" is "if \(Q\) then \(P\)". You do not negate anything — that would give the inverse or contrapositive.
Does a true statement always have a true converse?
No. "If \(n\) is a multiple of \(4\), then \(n\) is even" is true, but its converse "if \(n\) is even, then \(n\) is a multiple of \(4\)" is false, because \(6\) is even and not a multiple of \(4\).
When are two statements equivalent?
When each implies the other, so \(P\Leftrightarrow Q\) holds. Equivalently, they always have the same truth value. A statement is also equivalent to its contrapositive.
What do the symbols \(\Rightarrow\), \(\Leftrightarrow\) and \(=\) mean?
\(\Rightarrow\) means "implies" (one direction), \(\Leftrightarrow\) means "is equivalent to" or "if and only if" (both directions), and \(=\) links two values, such as \(x=2\), not two statements.
What does "if and only if" mean?
"\(P\) if and only if \(Q\)" is the biconditional \(P\Leftrightarrow Q\). "\(P\) if \(Q\)" is \(Q\Rightarrow P\) and "\(P\) only if \(Q\)" is \(P\Rightarrow Q\); together they give the equivalence.