Mathematical Reasoning

The One Thing

Mathematical reasoning evaluates definite statements and combines them using logical connectors, quantifiers, and equivalence rules. The truth value of a compound statement depends on the truth values of its components and on the connector used.

Definitions and Results

• Statement: A declarative sentence that is either definitely true or definitely false, but not both. Questions, commands, exclamations, and sentences with unclear truth values are not statements.
• Truth Value: The value assigned to a statement: True (T) or False (F).
• Open Sentence: A sentence containing a variable whose truth value cannot be decided until a value is assigned to the variable.
• Negation: The denial of a statement, written as not $p$ or $\sim p$. If $p$ is true, $\sim p$ is false, and vice versa.
• Conjunction: $p$ and $q$, written $p\wedge q$. It is true only when both $p$ and $q$ are true; therefore, its only true case is $T\wedge T$.
• Disjunction: $p$ or $q$, written $p\vee q$. In mathematics, “or” is generally inclusive, so it is true when one or both component statements are true. It is false only for $F\vee F$.
• Implication: If $p$, then $q$, written $p\to q$. It is false only when $p$ is true and $q$ is false; it is true in all other cases. It does not assert that $p$ is true.
• Converse: For $p\to q$, the converse is $q\to p$. A statement and its converse may have different truth values.
• Inverse: For $p\to q$, the inverse is $\sim p\to \sim q$.
• Contrapositive: For $p\to q$, the contrapositive is $\sim q\to \sim p$. An implication and its contrapositive are logically equivalent.
• Biconditional: $p$ if and only if $q$, written $p\leftrightarrow q$. It is true when $p$ and $q$ have the same truth value: $T,T$ or $F,F$. It means that both $p\to q$ and $q\to p$ are true.
• Necessary Condition: $q$ is necessary for $p$ when $p$ cannot be true unless $q$ is true. This is represented by $p\to q$.
• Sufficient Condition: $p$ is sufficient for $q$ when $p$ being true guarantees that $q$ is true. This is represented by $p\to q$.
• Tautology: A compound statement that is true for every possible combination of truth values.
• Contradiction: A compound statement that is false for every possible combination of truth values.
• Logical Equivalence: Two statements are logically equivalent when they have the same truth value in every possible case, written $p\equiv q$.
• Quantifier: A word indicating how many elements satisfy a condition, such as “for all” $(\forall )$ or “there exists” $(\exists )$.
• Universal Statement: A statement asserting that a property is true for every member of a specified group, often using “all” or “for every.”
• Existential Statement: A statement asserting that a property is true for at least one member of a specified group, often using “there exists” or “some.”
• Counterexample: One example that proves a universal statement is false.

Important equivalences are:

$$\sim (p\wedge q)\equiv \sim p\vee \sim q$$

$$\sim (p\vee q)\equiv \sim p\wedge \sim q$$

$$p\to q\equiv \sim p\vee q$$

$$p\leftrightarrow q\equiv (p\to q)\wedge (q\to p)$$

For quantified statements:

$$\sim (\forall xP(x))\equiv \exists x\sim P(x)$$

$$\sim (\exists xP(x))\equiv \forall x\sim P(x)$$

Worked Methods

1. Determining whether a sentence is a statement

• Check that the sentence is declarative.
• Determine whether it has one definite truth value.
• Classify it as true or false.
• Exclude questions, commands, exclamations, and sentences whose truth value is unclear.
• Treat a sentence containing an unresolved variable as an open sentence until a value is assigned.

2. Evaluating a negation

• Identify the original statement $p$.
• Write its denial as $\sim p$.
• Reverse the truth value:

- If $p$ is $T$, then $\sim p$ is $F$. - If $p$ is $F$, then $\sim p$ is $T$.

• If the statement contains “and,” “or,” “all,” or “some,” apply the appropriate logical or quantifier rule rather than merely inserting the word “not.”

3. Evaluating a conjunction

• Identify $p$ and $q$.
• Form $p\wedge q$.
• Check both component truth values.
• Assign $T$ only when both are true; otherwise assign $F$.

For example, let $p$ mean “7 is prime” and $q$ mean “7 has exactly two positive factors.” Both $p$ and $q$ are true, so $p\wedge q$ is true.

4. Evaluating a disjunction

• Identify $p$ and $q$.
• Form $p\vee q$.
• Treat “or” as inclusive.
• Assign $F$ only when both $p$ and $q$ are false.

5. Evaluating an implication

• Identify the antecedent $p$ and consequent $q$.
• Form $p\to q$, meaning “if $p$, then $q$.”
• Check the four possible truth-value combinations.
• Assign $F$ only to $T\to F$; all other cases are true.
• Remember that $p\to q$ states only that whenever $p$ is true, $q$ must also be true.

Example:

• “If a number is divisible by 4, then it is even” is true.
• Its converse, “If a number is even, then it is divisible by 4,” is false.

6. Forming related implications

Given $p\to q$:

• Write the converse as $q\to p$.
• Write the inverse as $\sim p\to \sim q$.
• Write the contrapositive as $\sim q\to \sim p$.
• Use the equivalences:

- $p\to q$ is logically equivalent to $\sim q\to \sim p$. - $q\to p$ and $\sim p\to \sim q$ are logically equivalent to each other. - The converse and inverse are generally not equivalent to the original implication.

7. Evaluating a biconditional

• Identify $p$ and $q$.
• Form $p\leftrightarrow q$.
• Check whether $p$ and $q$ have the same truth value.
• Alternatively, evaluate both $p\to q$ and $q\to p$.
• The biconditional is true only when both implications are true.

For the example in which $p$ means “7 is prime” and $q$ means “7 has exactly two positive factors,” both statements are true. Therefore, $p\leftrightarrow q$ is true.

8. Testing a compound statement with a truth table

• Identify each component statement.
• List every possible combination of their truth values.
• Apply the relevant connector in each row.
• Determine whether the compound statement is always true, always false, or variable.
• Classify it as a tautology, contradiction, or neither where appropriate.

9. Negating compound and quantified statements

For a compound statement:

• Negate each component as required.
• Change the connector according to De Morgan’s laws:

- $\sim (p\wedge q)\equiv \sim p\vee \sim q$ - $\sim (p\vee q)\equiv \sim p\wedge \sim q$

For quantified statements:

• Change “for all” $(\forall )$ to “there exists” $(\exists )$, and negate the predicate:

$$\sim (\forall xP(x))\equiv \exists x\sim P(x)$$

• Change “there exists” $(\exists )$ to “for all” $(\forall )$, and negate the predicate:

$$\sim (\exists xP(x))\equiv \forall x\sim P(x)$$

10. Disproving a universal statement

• Identify the universal claim.
• Search for a member of the specified group that fails the property.
• Give one valid counterexample.
• Conclude that the universal statement is false.

Where It Goes Wrong

• Treating a question, command, exclamation, or unclear sentence as a statement, despite its lack of a definite truth value.
• Forgetting that $p\to q$ is false only when $p$ is true and $q$ is false; an implication does not assert that $p$ is true.
• Interpreting mathematical “or” as exclusive rather than inclusive, even though $p\vee q$ is true when both components are true.
• Assuming that the converse $q\to p$ has the same truth value as $p\to q$; the example involving divisibility by 4 shows that this is generally false.
• Negating a conjunction or disjunction without changing the connector, contrary to De Morgan’s laws.
• Forgetting to reverse the quantifier when negating a universal or existential statement.
• Attempting to disprove a universal statement through general discussion instead of providing one valid counterexample.
• Confusing necessary and sufficient conditions: $p\to q$ means $p$ is sufficient for $q$, while $q$ is necessary for $p$.

What Gets Asked

This material supports questions requiring students to:

• Decide whether a sentence is a statement, an open sentence, or neither.
• Assign truth values to individual and compound statements.
• Construct or complete truth tables for $\sim p$, $p\wedge q$, $p\vee q$, $p\to q$, and $p\leftrightarrow q$.
• Negate statements involving “and,” “or,” “all,” and “some.”
• Apply De Morgan’s laws and the quantified negation rules.
• Find the converse, inverse, and contrapositive of an implication.
• Identify which forms are logically equivalent.
• Distinguish necessary from sufficient conditions.
• Express an implication as a disjunction and a biconditional as two implications.
• Classify compound statements as tautologies, contradictions, or neither.
• Use a counterexample to disprove a universal statement.
• Evaluate the examples “7 is prime” and “7 has exactly two positive factors,” including $p\wedge q$ and $p\leftrightarrow q$.
• Assess the implication “If a number is divisible by 4, then it is even,” its converse, and the differing truth values of the two statements.