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⇎ Negated Biconditional_
The negated biconditional, written p ⇎ q, asserts that p and q are not equivalent: they have different truth values. Its table is identical to exclusive disjunction ⊕, so p ⇎ q ≡ ¬(p ⇔ q) ≡ p ⊕ q. It is the natural tool for spotting the rows where two formulas differ.
Truth table: p ⇎ q
| p | q | p ⇎ q★ |
|---|---|---|
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Classification: Contingency · 4 rows
Definition
p ⇎ q is true when p and q have different values and false when they agree. It is the table of p ⇔ q with every value flipped and, at the same time, the very same table as p ⊕ q.
Textbooks usually write ¬(p ⇔ q), ¬(p ↔ q) or p ≢ q. The calculator includes the symbol ⇎ so the negation appears as a single step.
How to read it
p ⇎ q is read "p is not equivalent to q", "not (p if and only if q)", or "p and q differ". It describes the situation where exactly one of the two propositions is true.
When it is true
The table has four rows. With p = T and q = T, p ⇎ q = F. With p = T and q = F, p ⇎ q = T. With p = F and q = T, p ⇎ q = T. With p = F and q = F, p ⇎ q = F.
Two true rows and two false rows: a contingency with exactly the XOR table.
Everyday example
To check whether two formulas A and B are equivalent, build the table of A ⇎ B. Any T that appears marks a counterexample: that row is where the formulas differ. If the whole column is F, the formulas are equivalent. This is the method the calculator uses to compare expressions.
Properties and equivalences
Definition: p ⇎ q ≡ ¬(p ⇔ q). Equivalence with XOR: p ⇎ q ≡ p ⊕ q. Normal form: p ⇎ q ≡ (p ∧ ¬q) ∨ (¬p ∧ q).
It inherits XOR's properties: it is commutative and associative, p ⇎ p is a contradiction, and p ⇎ ¬p is a tautology.
Common mistakes
Thinking that negating "p if and only if q" yields "p if and only if not q" and that this is something different. In fact ¬(p ⇔ q) ≡ p ⇔ ¬q ≡ p ⊕ q: all three have the same table.
Confusing ⇎ with ⇏. The former negates the equivalence (two true rows); the latter negates the one-way implication (one true row).
Equivalent expressions
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related laws and rules
Frequently asked questions
Is p ⇎ q the same as XOR? ▼
Yes. They have exactly the same truth table: true when p and q differ, false when they agree. p ⇎ q ≡ p ⊕ q.
What is the negated biconditional useful for? ▼
For comparing formulas: A ⇎ B is true precisely in the rows where A and B give different results. If its column is all false, A and B are equivalent.
What is the difference between ⇎ and ⇏? ▼
⇎ negates the two-way equivalence and has two true rows. ⇏ negates the one-way implication and has a single true row (p = T, q = F).