1
Equivalence is agreement in every row
Write the two columns side by side. If any row has a T in one and an F in the other, the sentences are not equivalent, and that row is the proof.
Lesson · Arguments and Validity · Propositional logic
Equivalence, De Morgan and Contraposition
Two sentences are equivalent when their columns agree in every row. De Morgan's laws turn “not both” into “at least one not” and “neither” into “both not”. The contrapositive of a conditional is equivalent to it; the converse and the inverse are not.
equivalent: the columns agree in every row
¬(p ∧ q) ≡ ¬p ∨ ¬q not both
¬(p ∨ q) ≡ ¬p ∧ ¬q neither
p → q ≡ ¬q → ¬p contrapositive
not equivalent: q → p, ¬p → ¬q
Three equivalences, two impostors
Core ideas
What this lesson establishes
Two sentences that agree in every row can be swapped in any argument without changing its verdict. That makes equivalence worth checking.
2
Negation flips the connective
To negate a conjunction, negate the parts and switch to or; to negate a disjunction, negate the parts and switch to and. Negating the parts alone gets the connective wrong.
3
A conditional has one equivalent turn
From p → q there are three rewritings. Only the contrapositive ¬q → ¬p keeps the table. The converse q → p and the inverse ¬p → ¬q change it.
The material
Same columns, same sentence
Three equivalences to know, each checked by a table, and the near misses that look like them.
Definition · Equivalence
Two formulas are equivalent, written A ≡ B, when they have the same truth value in every row. One row with different values shows they are not.
Theorem · De Morgan
¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q. In words: “not both” is “at least one is not”, and “neither” is “both are not”.
p q | ¬(p ∧ q) ¬p ∨ ¬q ¬p ∧ ¬q ¬(p ∨ q)
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T T | F F F F
T F | T T F F
F T | T T F F
F F | T T T T
Compare the columns. The first two agree in all four rows, which is the first law. The last two agree in all four, which is the second. The first and third do not agree: they differ in the second and third rows, which is why “not both” and “neither” are different claims. Not both is satisfied when exactly one holds; neither is not.
The contrapositive is a second, separate equivalence. For a conditional p → q, the contrapositive is ¬q → ¬p: swap the parts and negate each. The converse swaps only (q → p) and the inverse negates only (¬p → ¬q). The lab sets the conditional against all three.
p q | p → q ¬q → ¬p q → p ¬p → ¬q
---------------------------------------------
T T | T T T T
T F | F F T T
F T | T T F F
F F | T T T T
The contrapositive column is the conditional's column. The converse and inverse columns are the same as each other, and differ from the conditional in the second and third rows. So a conditional and its contrapositive stand or fall together, which is why a proof of one is a proof of the other, and why modus tollens is valid.
Example · A sentence and its contrapositive
“If a shape is a square, it has four sides” and “if a shape does not have four sides, it is not a square” say the same thing. The converse, “if it has four sides it is a square”, is false of a rectangle that is not a square.
Interactive
Truth table
Every assignment, evaluated
Method
Deciding whether two sentences are equivalent
The test is the same every time; only the formulas change.
1
Put both formulas in the same table
Use the same letters in the same order, so that every row means the same case for both.
2
Compare the last columns row by row
Look for any row where one says T and the other F. You are looking for one disagreement, and need no more.
3
If you find one, the sentences are not equivalent
Report the row. It is an assignment under which one sentence is true and the other false, and it is the whole proof.
4
If you find none, they are equivalent
All rows agree. Either can replace the other in any argument.
Equivalent or not
Score 0 / 0Choose the answer you can justify.
Rich and famous
“It is not the case that she is both rich and famous.” Let r be “she is rich” and f be “she is famous”.
r f | ¬(r ∧ f) ¬r ∨ ¬f ¬r ∧ ¬f
T T | F F F
T F | T T F
F T | T T F
F F | T T T
The first two columns agree in all four rows, so “not both rich and famous” is “not rich or not famous”. The third column does not: in the second row she is rich but not famous, the sentence is true, and “she is neither” is false.
Common mistakes
1
Writing ¬(p ∧ q) as ¬p ∧ ¬qNegating the parts is not enough: the connective flips. In the row where p is true and q false, ¬(p ∧ q) is true and ¬p ∧ ¬q is false. The correct partner of “not both” is ¬p ∨ ¬q, whose column the lab shows agreeing in all four rows.
2
Offering the converse or inverse as the contrapositiveOf the three rewritings of p → q, only ¬q → ¬p has the same table. The converse and inverse each differ from it in two rows. “If it is a square it has four sides” is true and its converse is not.
3
Treating “not both” and “neither” as one claim¬(p ∧ q) is true in three rows and ¬(p ∨ q) in one. They differ when exactly one of p, q is true. A claim that she is not both rich and famous is made true by her being rich only.
Check your understanding
✓
Finish when you can negate and rewrite with a reason.
Given two formulas, decide whether they are equivalent by comparing columns and name a separating row if not; negate a conjunction or a disjunction by De Morgan; and write a conditional's contrapositive without producing its converse or its inverse.
Note: Equivalent sentences are interchangeable inside an argument without changing its verdict. That is how a long sentence is replaced by a short one in the lessons ahead, and how an invalid-looking argument is recognised as modus tollens.