Common Mistakes

Contents

• Logic
• Proof techniques
• Sequences

Logic

Truth tables

• Missing intermediate columns
  A complete truth table needs to include intermediate steps/columns. For example, if the final formula is $\sim (p \land \sim q) \rightarrow (q \lor r)$, then you should include intermediate columns such as $\sim q$, $(p \land \sim q)$, $\sim(p \land \sim q)$, $(q \lor r)$ before computing the final column.
• Incorrect precedence for logical connectives
  Another common issue is forgetting operator precedence. For example, some people treat $\sim p \lor q \rightarrow r$ as $\sim p \lor (q \rightarrow r)$, but it should be $(\sim p \lor q) \rightarrow r$, since $\land$ and $\lor$ have higher precedence than conditionals.
• Not writing conclusion
  For problems where you use truth tables to show equivalence, tautology, or contradiction, your conclusion should be explicit and complete. For example:
  • Equivalent: The truth table shows that $(\dots)$ and $(\dots)$ always have the same truth values, therefore they are equivalent.
  • Not equivalent: The truth table shows that $(\dots)$ and $(\dots)$ have different truth values in row(s) $(\dots)$, therefore they are not equivalent.
  • Tautology: Since all truth values of $(\dots)$ are $T$, $(\dots)$ is a tautology.
  • Not tautology: Since the truth value of $(\dots)$ is $F$ in row(s) $(\dots)$, $(\dots)$ is not a tautology.
  Similarly for contradiction, and valid arguments.
• Shuffling rows
  Do not shuffle rows. Keep the same order of writing the truth table rows -- TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
• Writing 1 and 0 in place of T and F
  Do not write 1 and 0 in place of T and F in truth tables. Similarly, do not write T and F in place of 1 and 0 in I/O tables.

Negations

• Negating compound inequalities
  To negate a compound statement such as $50 < x < 75$, first rewrite it as $(50 < x) \land (x < 75)$, then use De Morgan's law to negate. For example, writing $50 \ge x \ge 75$ is not a correct negation.
• Statements with conjunctions and disjunctions
  Statements like $p \land q \lor r$ can be confusing to read, because they could be interpreted as $(p \land q) \lor r$ or $p \land (q \lor r)$. So when writing these, group the “or” using parentheses, e.g. “$n$ is divisible by 6 and ($n$ is not divisible by 2 or $n$ is not divisible by 3).”
• Interchange formal and informal statements
  If the question is in formal language, write the negation in formal language and vice versa.

Arguments

• Not highlighting critical rows
  Critical rows should be highlighted in the truth tables that are used to verify the validity and invalidty of arguments. If there are false conclusions in critical rows, they should be highlighted as well.
• Not following the organized format
  For problems involving proofs using valid arguments, the format used in the lecture notes / book should be used.
• Missing reasons/justifications
  Write proofs as a sequence of short steps, and give a reason for each step (rule, law, definition, or an earlier line).