Textbook
Equivalence $P\Leftrightarrow Q$ requires both implications: $P\Rightarrow Q$ and $Q\Rightarrow P$. The notation means “if and only if.” Each condition is necessary and sufficient for the other. Their truth values agree: both true or both false.
For a real number $x$, the identity is $x^2=0\Leftrightarrow x=0$. A proof needs both directions: only zero has square zero, and zero squared is zero. Proving one direction licenses $\Rightarrow$, not $\Leftrightarrow$.
Matching truth values at one point establishes equivalence only for that particular pair of propositions. Replacing conditions $P(x)$ and $Q(x)$ throughout a domain requires justification for every permitted $x$. One input giving different answers refutes the replacement. Two false answers are entirely compatible with equivalence: it compares truth values rather than requiring each side to be true.
Definitions and further reading: Mathematics for Computer Science.
In plain language
Equivalent conditions admit exactly the same candidates, so either can replace the other without changing the result. Their answers must agree throughout the domain, including cases where both are false. A one-way turnstile comparison does not rule out extra visitors at the second, while a shared refusal to one person does not establish identical rules. That is why “divisible by 4” cannot replace “even”: the passenger-exchange agreement fails on other numbers.
Example
The example compares $P(x):x^2=0$ and $Q(x):x=0$ at the illustrative values −1, 0, and 1. This example differs from proving both directions for arbitrary real $x$.
Equivalence: Both Directions at Once
Checking values
Both conditions admit only zero.
Step 1 of 3: Checking values
Read all steps
Checking values
Both conditions admit only zero.
- State
- −1: false/falsecurrent0: true/truecurrent1: false/falsecurrent
Forward direction
If the square is zero, the number is zero.
- State
- x² = 0currentx = 0current
Reverse direction
Substituting zero gives square zero; both deductions are needed.
- State
- x = 0currentx² = 0current
At $x=-1$ and $x=1$, both checks are false while their equivalence is true. At zero, both checks are true and the equivalence remains true. The table illustrates agreement; the general proof uses the real-number property that a nonzero number has a positive square. The reverse direction follows by substituting zero. Neither arrow can be skipped merely because a finite table looks convincing.
Related concepts: Implication.