2.1 Logic Statements, Conditionals, and Truth

Reading Logic Statements Precisely

Logical reasoning is a smaller domain on the Mathematics 30-2 Diploma Examination than functions or probability, but the 2025-2026 information bulletin still assigns it 15-20% of the exam. It also connects strongly to written-response communication: a correct calculation can lose force if the reasoning statement beside it is too vague. In this course, logic usually appears through statements about games, puzzles, sets, number properties, or conditions in a real situation.

A statement is a sentence that can be judged true or false. The claim "12 is divisible by 3" is a statement. The instruction "choose a number" is not a statement because it is a command. The sentence "that puzzle is hard" is not useful as a logic statement unless the exam defines a measurable meaning for hard.

Conditional statements

A conditional statement has the form "if p, then q." The part after if is the hypothesis. The part after then is the conclusion.

The conditional and the contrapositive always have the same truth value. The converse and inverse also match each other, but they do not have to match the original conditional. In the table, the conditional is true because every multiple of 6 is even. The converse is false because 8 is even but not a multiple of 6. That single example is enough to reject the reversed claim.

Truth and allowed conclusions

A common diploma trap is treating a rule as if it works both ways. Suppose a board-game rule says: "If a token lands on a shaded square, then the player draws a card." You are told that Maya drew a card. You cannot conclude that Maya landed on a shaded square, because there may be other card-drawing rules. You are told that Leo did not draw a card. You can conclude that Leo did not land on a shaded square, because the contrapositive follows from the original rule.

This distinction is easier if you make a two-row evidence check:

1. Is the condition in the original if part confirmed? If yes, the conclusion follows.
2. Is the conclusion in the original then part denied? If yes, the condition is denied.
3. Is only the conclusion confirmed or only the condition denied? Then the original rule does not settle the case.

Negation and quantifiers

The negation of a statement says exactly that the original is false. Negations become subtle when the statement uses all, some, none, always, or never.

Do not negate "greater than 7" as "less than 7"; equality has to go somewhere. Do not negate "all" as "none"; one counterexample is enough to make an all statement false.

Worked diploma-style reasoning

Consider this claim: "If a student solved every set problem correctly, then the student drew a Venn diagram." Four student records are given.

• Ari solved every set problem correctly and drew a Venn diagram.
• Blake solved every set problem correctly and did not draw a Venn diagram.
• Chen did not solve every set problem correctly and drew a Venn diagram.
• Dina did not solve every set problem correctly and did not draw a Venn diagram.

Only Blake makes the conditional false. A conditional fails when the hypothesis is true and the conclusion is false. Chen does not disprove the claim because the hypothesis was not met. Dina also does not disprove the claim for the same reason. Ari supports the claim, but support from one example is not proof that the rule always holds.

How to show logic in written response

When logic appears in written response, write the rule and the case separately. For example: "All shaded-square landings require a card draw. Leo did not draw a card. Therefore Leo could not have landed on a shaded square." That is stronger than writing "Leo did not draw a card, so no shaded square" with no link between the rule and the conclusion.

The safest format is:

• Define the condition and result in words.
• State whether the original rule, converse, or contrapositive is being used.
• Name a counterexample when rejecting a universal claim.
• Avoid extra assumptions about unstated rules.

Diploma traps to avoid

The word or in mathematics usually means inclusive or: A or B means A, B, or both unless the context says exactly one. The word and means both conditions at the same time. The phrase if and only if means both the conditional and its converse are true, which is much stronger than a one-way rule.

When a question asks for a conclusion that must be true, do not choose a statement that is merely possible. When it asks which statement is false, test the most restrictive words first: all, none, must, always, and never. Those words are easy to break with one valid case.