Sufficient: enough on its own
Read "if n is a multiple of 4, then n is even" as a statement about what is enough. Once you know that n is a multiple of 4, you know that n is even, and nothing else needs checking. Being a multiple of 4 is a sufficient condition for being even.
In general, when is true, P is sufficient for Q: P on its own guarantees Q.
Necessary: cannot be done without
Now read the same statement from its other end. A number cannot be a multiple of 4 without being even, so being even is a necessary condition for being a multiple of 4. It is required, but it is not enough.
In general, when is true, Q is necessary for P. The contrapositive says why: if not Q then not P, so without Q there is no P.
One arrow, read both ways
Sufficient means enough on its own. Necessary means it cannot be done without. One true conditional, , says both at once: P is sufficient for Q, and Q is necessary for P.
The usual trap is to turn the second reading around. "P is sufficient for Q" means . "P is necessary for Q" means , because Q cannot happen without P.
With the example, both sentences describe the same arrow. "Being a multiple of 4 is sufficient for being even" means if n is a multiple of 4, then n is even. "Being even is necessary for being a multiple of 4" also means if n is a multiple of 4, then n is even. It does not mean that every even number is a multiple of 4.
a statement and its contrapositive are broken by the same numbers, which is why one is true whenever the other is
Find a number that makes the converse false
The gold circle, the multiples of 4, lies inside the circle of even numbers. Every number in the gold circle is in the even circle, so being a multiple of 4 is sufficient for being even. A number outside the even circle is outside the gold one too, so being even is necessary. Move the number being tested into the ring between the circles: there, at n = 6, a number is even without being a multiple of 4, so being even is not sufficient.
Necessary without being sufficient
A condition can be necessary without being sufficient. Being even is necessary for being a multiple of 4, but 6 is even and is not a multiple of 4, so being even is not sufficient. In the same way, having four sides is necessary for being a square but not sufficient, since a rectangle that is not a square also has four sides.
The other way round, being a multiple of 4 is sufficient for being even but not necessary: 6 is even without being a multiple of 4.
When a condition is both necessary and sufficient, each statement forces the other. A triangle has three equal sides exactly when it has three equal angles, so each condition is necessary and sufficient for the other. Such a pair is written with "if and only if".
The words “if” and “only if”
The same two ideas hide in two short phrases. "P if Q" means , so Q is sufficient for P. "P only if Q" means , so Q is necessary for P: P happens only when Q does.
The usual mistakes
Turning necessary around. "Being even is necessary for being a multiple of 4" means every multiple of 4 is even. It does not mean every even number is a multiple of 4.
Treating a necessary condition as enough. Six is even, and that does not make it a multiple of 4.
Reading a decision into a condition that gives none. When a sufficient condition fails, or a necessary condition holds, the statement makes no promise either way.
Steel rods and a length check
In the application below, a rod passes a length check when its length L mm satisfies . Every passing rod has , so that condition is necessary. Every length with passes, so that narrower condition is sufficient.
Worked example: Steel Rods Checked Against a Length Tolerance, and What a Shipping Rule Decides
Question A factory cuts steel rods to a target length of 200 mm. A rod passes the length check exactly when its length L mm satisfies 199.5 ≤ L ≤ 200.5. (a) For passing the length check, is the condition L ≥ 199.5 necessary, sufficient, both or neither? Is the condition 199.8 ≤ L ≤ 200.2 necessary, sufficient, both or neither? (b) The factory's shipping rule says: "Passing the length check is necessary for a rod to be shipped, and passing both the length check and the straightness check is sufficient." Rod X is 200.6 mm long and passes the straightness check. Rod Y is 200.1 mm long and fails the straightness check. Using only this rule, is each rod shipped, not shipped, or left undecided?
1.Every rod that passes has L ≥ 199.5, so the condition L ≥ 199.5 is necessary. A rod of 201 mm meets it and still fails the check, so it is not sufficient.
Every passing rod has L ≥ 199.5, but a 201 mm rod meets it and fails. 2.Every length with 199.8 ≤ L ≤ 200.2 lies inside 199.5 ≤ L ≤ 200.5, so that condition is sufficient. A rod of 200.4 mm passes without meeting it, so it is not necessary.
Every length from 199.8 to 200.2 passes, but a 200.4 mm rod passes without meeting it. 3.(a) L ≥ 199.5 is necessary but not sufficient, and 199.8 ≤ L ≤ 200.2 is sufficient but not necessary.
(a) L ≥ 199.5 is necessary only, and 199.8 ≤ L ≤ 200.2 is sufficient only. 4."Passing the length check is necessary for shipping" means: if a rod is shipped, it passed the length check. Its contrapositive is: if a rod fails the length check, it is not shipped. Rod X is 200.6 mm, which is more than 200.5, so it fails the length check and is not shipped.
Rod X fails the necessary condition, so it is not shipped. 5.(b) Rod Y passes the length check, but a necessary condition that holds promises nothing. It fails the straightness check, so the sufficient condition does not hold, and a sufficient condition that fails says nothing either. Rod X is not shipped, and the rule leaves Rod Y undecided.
(b) Rod Y meets the necessary condition and misses the sufficient one, so the rule does not decide it.
Answer: (a) L ≥ 199.5 is necessary but not sufficient; 199.8 ≤ L ≤ 200.2 is sufficient but not necessary. (b) Rod X is not shipped; the rule leaves Rod Y undecided.
Common mistakes
- Calling L ≥ 199.5 sufficient because every passing rod meets it. That shows the condition is necessary. Sufficient is the other direction, and the 201 mm rod meets the condition and fails.
- Saying Rod Y is not shipped because it fails the straightness check. The rule makes straightness part of a sufficient condition, not a necessary one, so failing it leaves the rule silent about Rod Y.
More mathematical statements problems, worked step by step →