Math101 · Independent Ontario learning library
Math101learn.math101.caLogic and Statements
A careful foundation in propositions, connectives, implication, and the difference between syntax and truth.
Precise definition
A proposition is a declarative sentence with a definite truth value, true or false. From propositions $p$ and $q$, negation $\neg p$, conjunction $p\land q$, disjunction $p\lor q$, implication $p\to q$, and biconditional $p\leftrightarrow q$ form compound propositions. In standard inclusive logic, $p\lor q$ allows both to be true.
Notation and mathematical language
The implication $p\to q$ is false only when $p$ is true and $q$ is false. Its converse is $q\to p$, inverse is $\neg p\to\neg q$, and contrapositive is $\neg q\to\neg p$. Only the contrapositive is logically equivalent to the original implication in general.
Conceptual picture
Logical form separates a pattern of reasoning from a sentence's subject matter. Vacuous truth may feel unfamiliar: if $p$ is false, no counterexample of the form 'true premise, false conclusion' occurs, so $p\to q$ is true. This convention makes implication compatible with proof and set inclusion.
Conditions and key results
A sentence with an unassigned variable, such as $x>3$, is a predicate rather than a proposition until a value or quantifier fixes its truth. Natural-language words such as 'or', 'unless', 'only if', and 'necessary' must be translated carefully. $p$ only if $q$ means $p\to q$.
A reliable strategy
- Identify atomic propositions and assign one symbol to each without changing their meanings.
- Translate connectives, paying special attention to implication direction and inclusive versus exclusive or.
- Use definitions or a truth table to evaluate or compare the compound statement.
- Translate the result back into clear language and test a counterexample-shaped case.
Fully worked example
Interpretation and application
Formal logic supports proof, database queries, digital circuits, and software conditions. In a program, confusing 'necessary' with 'sufficient' can admit invalid input or reject valid input; writing the intended implication before coding reduces that risk.
Common mistakes
Verification and reasonableness
- Construct the one potential falsifying case for an implication: true premise and false conclusion.
- Compare a proposed reformulation across all truth assignments.
- Return to the original sentences and check that the direction of 'if', 'only if', necessity, and sufficiency is preserved.
Practice
- When is $p\to q$ false?
- What is the contrapositive of 'if it rains, the field is wet'?
- Is $x+1=4$ a proposition before $x$ is specified?
Answers and brief solutions
- Exactly when $p$ is true and $q$ is false.
- If the field is not wet, then it did not rain.
- No. It is an open sentence or predicate.
Further deduction
Necessary and sufficient conditions encode implication direction. Saying '$P$ is sufficient for $Q$' means $P\to Q$; saying '$P$ is necessary for $Q$' means $Q\to P$. Saying '$P$ iff $Q$' asserts both. For example, divisibility by 4 is sufficient but not necessary for evenness, while evenness is necessary but not sufficient for divisibility by 4. Translating these phrases before reasoning prevents a common converse error.
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Using T=1 and F=0, what is the truth value of T implies F?
- The premise is true and the conclusion is false.
- Therefore $T\to F$ is false, represented by 0.
End of lesson
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