Worked Examples
Master the art of converting English to logic.
1Worked Example
Problem
Translate: "You can enter only if you have a ticket."
Let E = enter, T = have ticket.
Let E = enter, T = have ticket.
1
Identify "only if" — this creates an implication FROM the main action.
2
"A only if B" means A → B (B is necessary for A)
3
Answer:
E → T — If you enter, you have a ticket.⚠
Watch Out!
"Only if" ≠ "if". "A only if B" is A→B, but "A if B" is B→A. Opposite directions!
2Worked Example
Problem
Translate: "Unless you apologize, we're done."
Let A = apologize, D = done.
Let A = apologize, D = done.
1
"Unless" means "if not". Rewrite: "If you don't apologize, we're done."
2
Answer:
¬A → D3
Equivalently:
¬D → A (contrapositive) — If we're not done, you apologized.⚠
Watch Out!
"Unless" always introduces a negated condition in the antecedent.
3Worked Example
Problem
Translate: "Neither the manager nor the director approved the budget."
Let M = manager approved, D = director approved.
Let M = manager approved, D = director approved.
1
"Neither A nor B" means both are false.
2
Direct form:
¬M ∧ ¬D3
De Morgan equivalent:
¬(M ∨ D) — "It's not the case that either approved."✦Tip
Key patterns: "only if" → forward implication, "unless" → negated condition, "neither...nor" → both negated with AND.