Mathematical Reasoning — JEE Mains Mathematics

Formulas (3)

• The one row where an implication is false · When an implication is false
  $$A\rightarrow B\equiv\sim A\vee B,\quad\text{false only at }A=T,\ B=F$$
• Testing for a tautology · The tautology test
  $$A\rightarrow B\equiv\sim A\vee B,\qquad p\vee\sim p\equiv T,\qquad p\wedge\sim p\equiv F$$
• Judging two claims, (S1) and (S2) · A statement and its negation
  $$X\vee\sim X\equiv T,\qquad X\wedge\sim X\equiv F$$

Watch out for (3)

• A false antecedent makes it true
  → The one row where an implication is false
• Neither is a third answer
  → Testing for a tautology
• Judge the claim, not only the statement
  → Judging two claims, (S1) and (S2)

Formulas (2)

• Choosing the connective · The four connectives
  $$p\wedge q,\quad p\vee q,\quad p\rightarrow q\equiv\sim p\vee q,\quad p\leftrightarrow q\equiv(p\rightarrow q)\wedge(q\rightarrow p)$$
• Choosing the missing statement · Useful reductions
  $$p\wedge\sim p\equiv F,\qquad F\rightarrow X\equiv T,\qquad X\rightarrow T\equiv T$$

Watch out for (2)

• $\leftrightarrow$ is not $\rightarrow$
  → Choosing the connective
• Test every candidate
  → Choosing the missing statement

Formulas (3)

• Negating an implication · Negation of an implication
  $$\sim(A\rightarrow B)\equiv A\wedge\sim B$$
• De Morgan's laws · De Morgan's laws
  $$\sim(A\wedge B)\equiv\sim A\vee\sim B,\qquad\sim(A\vee B)\equiv\sim A\wedge\sim B$$
• Simplifying to a short equivalent · Absorption
  $$p\vee(p\wedge q)\equiv p,\qquad p\wedge(p\vee q)\equiv p$$

Watch out for (3)

• Not the converse, not the inverse
  → Negating an implication
• Flip the connective too
  → De Morgan's laws
• Check one row before choosing
  → Simplifying to a short equivalent

Formulas (2)

• Equivalent forms of an implication · Or-form of an implication
  $$A\rightarrow B\equiv\sim A\vee B\equiv\sim B\rightarrow\sim A$$
• Converse, contrapositive and statements in words · The related statements
  $$p\rightarrow q\equiv\sim q\rightarrow\sim p,\qquad q\rightarrow p\equiv\sim p\rightarrow\sim q$$

Watch out for (2)

• An 'or' of implications gives an 'and' on the left
  → Equivalent forms of an implication
• 'Only if' points forward
  → Converse, contrapositive and statements in words