Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{To prove an identity like }(x+1{)}^2-(x-1{)}^2\equiv 4x\text{, you should move terms across the equals sign (e.g., adding }(x-1{)}^2\text{ to both sides).} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Method: You should start with one side (usually the more complicated LHS) and manipulate it algebraically until it matches the other side.} \end{gathered}$$
• $$\begin{gathered} \text{In a mathematical proof involving integers, how should you algebraically represent:} \\ \text{1. An even number?} \\ \text{2. An odd number?} \end{gathered}$$
  $$\begin{gathered} \text{1. Even number: }2n\text{ (where }n\text{ is an integer)} \\ \text{2. Odd number: }2n+1\text{ (or }2n-1\text{)} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{Showing that a statement works for 5 different random examples proves that the statement is true for all numbers.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Concept: Examples can only be used for disproof (counter-examples). To prove a statement is true for *all* numbers, you must use algebraic deduction or proof by exhaustion (if the set of numbers is small/finite).} \end{gathered}$$
• $$\begin{gathered} \text{A question asks you to disprove the statement: } \\ \text{"For all real numbers }x\text{, }{x}^2>x\text{."} \\ \text{What is the most efficient method?} \end{gathered}$$
  $$\begin{gathered} \text{Disproof by Counter-example} \\ \text{Find one specific value where the statement fails (e.g., }x=0.5\text{, since }0.25\ngtr 0.5\text{).} \end{gathered}$$