Necessary and sufficient conditions
Tell when a condition is necessary, when it is sufficient and when it is both, and refute a false claim by a counter-example.
- Russia: grade 7
Guide
In the Russian programme the words "necessary" and "sufficient" enter gradually in Grades 7-9 with the signs and properties of figures (a congruence criterion is a sufficient condition, a property is a necessary one). This is advanced material: it is not in the basic Grade 7 course and is meant for olympiad preparation.
A is sufficient for B if A implies B ("if a number is divisible by 12 it is divisible by 6"). A is necessary for B if B cannot happen without A ("for a number to be divisible by 6 it must be divisible by 3"). If both hold we say "necessary and sufficient" ("if and only if").
To show a condition is not sufficient we find a counter-example: a number that meets the condition but not the claim (9 is divisible by 3 but not by 6). A common slip: confusing a statement with its converse.