Conditional and biconditional statements; tautologies and contradictions – Introduction to Logic (Senior 4 Mathematics) | Ganzaa

Introduction to Logic

Lessons

Senior 4 · Mathematics

Introduction to Logic

5 lessons

Lesson 2 of 5

Conditional and biconditional statements; tautologies and contradictions

25 min

When is a promise broken?

A teacher promises: "If you score at least 80% in the test, you will get a prize." Think of the four possible cases. You score 80% and get a prize: promise kept. You score 80% and get no prize: promise broken. You score less, and either do or do not get a prize: the teacher said nothing about that case, so the promise is not broken. A statement of the form "if $p$ , then $q$ " is false in exactly one case. Almost every theorem in mathematics has this "if … then" form, so this is the most important connective of all.

Definition

Conditional (implication) $p \Rightarrow q$ , "if $p$ then $q$ ": false only when $p$ is true and $q$ is false; true in every other case. $p$ is the hypothesis (antecedent) and $q$ the conclusion (consequent).

The same statement can be read as: " $p$ implies $q$ ", " $q$ if $p$ ", " $p$ only if $q$ ", " $p$ is a sufficient condition for $q$ ", " $q$ is a necessary condition for $p$ ".

Biconditional $p \Leftrightarrow q$ , " $p$ if and only if $q$ ": true when $p$ and $q$ have the same truth value. It means $(p \Rightarrow q) \wedge (q \Rightarrow p)$ : each is necessary and sufficient for the other.

$p$	$q$	$p \Rightarrow q$	$q \Rightarrow p$	$p \Leftrightarrow q$
T	T	T	T	T
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

The last two rows of $p \Rightarrow q$ surprise many students: a conditional with a false hypothesis is true. "If $2 + 2 = 5$ , then Kigali is in Rwanda" is a true statement, because the promise "if $2 + 2 = 5$ …" can never be broken. Such a conditional is said to be vacuously true.

Definition

Converse, inverse and contrapositive of the conditional $p \Rightarrow q$ :

converse: $q \Rightarrow p$ (swap the parts);
inverse: $\neg p \Rightarrow \neg q$ (negate both parts);
contrapositive: $\neg q \Rightarrow \neg p$ (swap and negate).

A truth table shows that a conditional always has the same truth value as its contrapositive, and the converse always has the same truth value as the inverse. A conditional and its converse can differ.

Worked example 1: the four related conditionals

For the statement "If an integer is divisible by 6, then it is divisible by 3", write the converse, inverse and contrapositive, and decide which are true.

Solution

Statement: true, since $n = 6 k = 3 (2 k)$ .
Converse: "If an integer is divisible by 3, then it is divisible by 6." False: 9 is divisible by 3 but not by 6.
Inverse: "If an integer is not divisible by 6, then it is not divisible by 3." False: again 9.
Contrapositive: "If an integer is not divisible by 3, then it is not divisible by 6." True, as it must be.

Warning

Do not reverse an implication. "If it rains, the ground is wet" does not give "if the ground is wet, it rained" (a pipe may have burst). Reasoning from $q$ back to $p$ is the commonest error in proofs. Use the contrapositive instead: "if the ground is dry, it did not rain" is safe.

Worked example 2: truth values

Find the truth value of each statement. (a) If $7$ is odd, then $7^{2}$ is even. (b) If $10 \lt 3$ , then $5$ is prime. (c) A triangle is equilateral if and only if its three angles are equal. (d) $4 \gt 1 \Leftrightarrow 4$ is odd.

Solution (a) T $\Rightarrow$ F, which is F ( $49$ is odd). (b) F $\Rightarrow$ T, which is T (vacuously). (c) Both directions are true, so T. (d) T $\Leftrightarrow$ F, which is F.

Definition

Tautology: a compound statement that is true in every row of its truth table, whatever the truth values of its parts. Example: $p \vee \neg p$ .

Contradiction: a compound statement that is false in every row. Example: $p \wedge \neg p$ .

Contingency: a compound statement that is neither: true in some rows and false in others. Example: $p \Rightarrow q$ .

Worked example 3: a tautology that is a rule of reasoning

Show that $[(p \Rightarrow q) \wedge p] \Rightarrow q$ is a tautology.

Solution

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \wedge p$	$[(p \Rightarrow q) \wedge p] \Rightarrow q$
T	T	T	T	T
T	F	F	F	T
F	T	T	F	T
F	F	T	F	T

Every entry of the last column is T, so it is a tautology. In words: "if $p$ implies $q$ , and $p$ is true, then $q$ is true". This rule, called modus ponens, is used in every proof; Lesson 5 returns to it.

Worked example 4: classifying statements

Classify (a) $(p \Rightarrow q) \Leftrightarrow (\neg p \vee q)$ (b) $(p \wedge q) \wedge \neg(p \vee q)$ (c) $(p \vee q) \Rightarrow p$ .

Solution

Rows TT, TF, FT, FF: $p \Rightarrow q$ is T, F, T, T and $\neg p \vee q$ is T, F, T, T. They always agree, so the biconditional is T in every row: a tautology. (So $p \Rightarrow q$ can always be rewritten as $\neg p \vee q$ .)
If $p \wedge q$ is true, then $p \vee q$ is true and $\neg(p \vee q)$ is false. So the conjunction is false in every row: a contradiction.
Row TT gives T, row FT gives T $\Rightarrow$ F, which is F: a contingency.

Tip

To show that $p \Rightarrow q$ is false, you need only one case where $p$ is true and $q$ is false: a counterexample. To show that it is true, it is often easier to prove the contrapositive $\neg q \Rightarrow \neg p$ .

The Organic Chemistry Tutor: Converse, Inverse and Contrapositive, conditional and biconditional statements. Watch for which of the related statements keep the truth value of the original.

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Practice

Use truth tables where needed.

Find the truth values: (a) If $3 \gt 5$ , then $2 + 2 = 4$ . (b) If 7 is odd, then 7 is even. (c) 4 is even $\Leftrightarrow$ 4 is a perfect square. (d) $10 \lt 3 \Leftrightarrow 2 \gt 1$ .
Write the converse, inverse and contrapositive of "If a quadrilateral is a square, then it has four right angles". Which are true?
Rewrite "A shape is a square only if it has four equal sides" in "if … then" form.
Construct the truth table of $p \Rightarrow \neg q$ .
Show that $p \Leftrightarrow q$ and $(p \Rightarrow q) \wedge (q \Rightarrow p)$ have the same truth table.
Classify as tautology, contradiction or contingency: (a) $p \Rightarrow (p \vee q)$ (b) $(p \vee q) \Rightarrow p$ (c) $\neg(p \Rightarrow q) \wedge q$ .
Show that $[(p \Rightarrow q) \wedge \neg q] \Rightarrow \neg p$ is a tautology.
$p \Rightarrow q$ is false. Find the truth values of $p$ , $q$ and $q \Rightarrow p$ .

Show answersHide answers

(a) F $\Rightarrow$ T: T (b) T $\Rightarrow$ F: F (c) T $\Leftrightarrow$ T: T (d) F $\Leftrightarrow$ T: F.
Converse: "If a quadrilateral has four right angles, it is a square": false (a 2 by 5 rectangle). Inverse: "If it is not a square, it does not have four right angles": false (same rectangle). Contrapositive: "If it does not have four right angles, it is not a square": true.
"If a shape is a square, then it has four equal sides."
Rows TT, TF, FT, FF: F, T, T, T.
Both columns are T, F, F, T.
(a) Tautology (if $p$ is true, so is $p \vee q$ ). (b) Contingency (F in row FT). (c) Contradiction: $\neg(p \Rightarrow q)$ is true only when $q$ is false, and then $\ldots \wedge q$ is false.
The last column is T in all four rows (this rule is called modus tollens).
$p$ is T and $q$ is F, so $q \Rightarrow p$ is F $\Rightarrow$ T, which is T.

Quick check

When is the conditional p ⇒ q false?

Key points

$p \Rightarrow q$ is false only when $p$ is true and $q$ is false.
$p \Rightarrow q$ has the same truth value as its contrapositive $\neg q \Rightarrow \neg p$ , but not always as its converse $q \Rightarrow p$ .
$p \Leftrightarrow q$ is true when $p$ and $q$ have the same truth value.
Tautology: always true; contradiction: always false; contingency: neither.

Sources and credits

Written by Ganzaa: the explanations, the diagram, the tables, the worked examples, the practice items and the questions.
Video: Converse, Inverse, & Contrapositive: Conditional & Biconditional Statements, Logic, Geometry, The Organic Chemistry Tutor, embedded from YouTube.

1When is a promise broken?

When is a promise broken?