# Implication and Equivalence

Mathematics I · Mathematical Language and Reasoning · https://tryals.app/learn/mathematics-i/implication-and-equivalence

## Implication and Truth

The **implication** (conditional) $p \to q$, read as "if $p$ then $q$," is the core connective of mathematical theorems. Here, $p$ is the **hypothesis** and $q$ is the **conclusion**.

An implication is **false in exactly one case**: when $p$ is true and $q$ is false. In all other cases it is true.

| Hypothesis ($p$) | Conclusion ($q$) | $p \to q$ | Truth Status |
| --- | --- | --- | --- |
| True | True | True | Normal |
| True | False | **False** | **The only failure** |
| False | True | True | **Vacuously true** |
| False | False | True | **Vacuously true** |

When $p$ is false, the implication is **vacuously true** regardless of $q$. A false premise never makes an implication false; only a true hypothesis paired with a false conclusion does.

## Derived Statements and Equivalence

From any implication $p \to q$, we form three related statements:

| Type | Form | Status |
| --- | --- | --- |
| **Converse** | $q \to p$ | **Not equivalent** |
| **Inverse** | $\neg p \to \neg q$ | **Not equivalent** |
| **Contrapositive** | $\neg q \to \neg p$ | **Logically equivalent** |

**Common pitfall:** Confusing an implication with its **converse** ($q \to p$) is a major error. $p \to q$ does **not** guarantee $q \to p$.

The **biconditional** $p \leftrightarrow q$, meaning "$p$ if and only if $q$" (iff), asserts that both $p$ and $q$ share the exact same truth value. It is true precisely when both $p \to q$ and its converse $q \to p$ hold simultaneously.

## Practice questions

4 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. A theorem's proof establishes p → q. A student concludes that q is true. What has gone wrong, and what would be needed?

A. q is true, but only for values of p that make the implication non-vacuous
B. The proof must have been of the contrapositive, which cannot be used this way
C. Nothing, proving an implication establishes its consequent
D. The implication only transfers truth if p is established

**Answer:** D. The implication only transfers truth if p is established

**Why:** A conditional promises q *given* p and says nothing on its own. Modus ponens needs both the implication and the premise, and forgetting the second is how a correctly proved theorem gets applied to a case it never covered, which is most of what goes wrong when a result is used outside its hypotheses.

Page: https://tryals.app/practice/mathematics-i/implication-and-equivalence/a-theorems-proof-establishes-p-q-a-student-concludes-that-q-is

### 2. "If n is divisible by 4 then n is even" is true, but its converse is false. Why does a true implication give no guarantee at all about its converse, while its contrapositive is automatically true?

A. Because the contrapositive is a restatement of the converse, so proving one proves the other.
B. Because converses are false by definition whenever the original implication is true.
C. Because p → q and ¬q → ¬p fail in exactly the same case.
D. Because the contrapositive reverses the direction of the implication, which preserves truth, while the converse negates it.

**Answer:** C. Because p → q and ¬q → ¬p fail in exactly the same case.

**Why:** Compare the failure conditions: p → q fails when p is true and q false, and ¬q → ¬p fails in precisely that same case, so the two stand or fall together. The converse fails in a different situation, and n = 6 supplies it. Converses are sometimes true, and when both hold you have an equivalence, which is what a characterisation theorem asserts.

Page: https://tryals.app/practice/mathematics-i/implication-and-equivalence/if-n-is-divisible-by-4-then-n-is-even-is-true-but-its-converse-is

### 3. Sort each conditional by its truth value. Use the real arithmetic facts.

**Answer:**

- True: If 2 + 2 = 4 then 3 is odd, If 2 + 2 = 5 then 3 is odd, If 2 + 2 = 5 then 1 = 2
- False: If 2 + 2 = 4 then 1 = 2

**Why:** Only c4 has a true hypothesis with a false conclusion, so only c4 is false. c2 and c3 are vacuously true because 2 + 2 = 5 is false, an implication with a false hypothesis is always true, however absurd its conclusion.

Page: https://tryals.app/practice/mathematics-i/implication-and-equivalence/sort-each-conditional-by-its-truth-value-use-the-real-arithmetic

### 4. Select every statement that is logically EQUIVALENT to 'if p then q'.

A. if q then p
B. if NOT p then NOT q
C. (NOT p) OR q
D. p AND (NOT q)
E. if NOT q then NOT p

**Answer:** C. (NOT p) OR q; E. if NOT q then NOT p

**Why:** The contrapositive 'if NOT q then NOT p' and the disjunctive form '(NOT p) OR q' both match the implication row for row. The converse and inverse are not equivalent, and 'p AND (NOT q)' is its NEGATION, the one case where the implication fails.

Page: https://tryals.app/practice/mathematics-i/implication-and-equivalence/select-every-statement-that-is-logically-equivalent-to-if-p-then-q