Necessary and Sufficient

The terms necessary and sufficient turn up a lot in mathematics. You might, for instance, have seen things like:

For two triangles to be congruent it is sufficient that they have two equal sides and the enclosed angle in common.

Or:

For two triangles to be similar it is necessary, but not sufficient, that they have an angle in common.

Or you might have seen parts of questions that say something like:

…by considering $(x^2 + gx + h)(x - k)$, or otherwise, show that $g^2 > 4h$ is a sufficient condition but not a necessary condition for the inequality

$$(g - k)^2 > 3(h - gk)$$

to hold

[STEP I 2001 question 3]

In this section, we shall explain how mathematicians use the term necessary and the term sufficient. We need to explain them as they have subtly different features from their everyday uses; the good thing is we have met the notions already, we just didn’t refer to them as necessity and sufficiency.

A is sufficient for B means exactly the same as if A then B. Usually we think of this as follows: A is sufficient for B if we can say that when A is true then we are guaranteed that B is true as well. And further, we need to note that if A is sufficient for B and we find that A is actually false, we cannot say whether B is true or false - as there might be cases where B is true and A is false.

The best way to think about A is sufficient for B is to think of it as saying that when we know A is true then we are guaranteed that B is true [and also remember that we cannot say anything about B when we are told A is false]

Here is an example:

$x$ is an odd natural number greater than 1 and not divisible by any natural number other than 1 and itself is sufficient for $x$ to be prime.

This is true because

if $x$ is an odd natural number greater than 1 and not divisible by any natural number other than 1 and itself then $x$ is prime.

is true. It is useful to note here that there is a case where $x$ is an odd natural number greater than 1 and not divisible by any number other than 1 and itself is false but $x$ is prime is true: i.e., the case $x = 2$. This is fine, though, as it does not make the statement itself false [check the truth table for if A then B] and, what is more, it illustrates the point we made above: there we said ‘there might be cases where B is true and A is false’ and here we have a case of this when $x = 2$.

Now necessity: in simple terms we say that A is necessary for B when if B then A, or equivalently A if B, is true. Usually, we think of this as follows: A is necessary for B if we can say that when B is true then we are guaranteed that A is true as well and if A is false then B must be false as well. And further, as before, we need to note that if A is necessary for B and we find that B is actually false, we cannot say whether A is true or false: there might be cases where A is true and B is false.

Here is an example of necessity:

two triangles having one side of the same length is necessary for the two triangles to be congruent.

We can see that this necessity condition is quite weak. It tells us something about congruence but not enough to guarantee that two triangles are congruent – two triangles each having one side of the same length is not sufficient to guarantee they are congruent!

We can now look at the term necessary and sufficient. From what we have written so far, it should be clear that if we write A is necessary and sufficient for B, then we mean A iff B. In other words, we mean that when A is true B is true, and vice versa, and when A is false then B is false, and vice versa.

Here is an example:

two triangles having the same three angles is a necessary and sufficient condition for the two triangles to be similar.

So, to summarise:

• if you are asked for a sufficient condition for B to be true then you need to look for a condition that guarantees to make B true.

• if you are asked for a necessary condition for B to be true then you need to look for something that must be the case for B to be true but might not be enough by itself to guarantee that B is true.

And if you are asked to find necessary and sufficient conditions for B then you need to look for something that guarantees the truth of B in all circumstances: that is, when your condition is true then B is true, and vice versa, and when your condition is false then B is false, and vice versa.

We can also think about necessity and sufficiency using a diagram. This diagram is slightly different from the diagrams we used earlier – although there are some connections between them – so it is best to look at these diagrams in isolation and treat them as a way of helping you grasp the notions of necessity and sufficiency.

Using the diagram, we can see that A is sufficient for B; that is to say, if we are inside the A circle then we must be inside the B circle too. Note that whilst A is sufficient for B, there are cases where we can be inside the B circle but outside the A circle; that is to say that even if A is false there is still the possibility that B is true – you should reconcile this with your formal understanding of A is sufficient for B, i.e. with if A then B which is also written as A ⟹ B

Now let’s look at necessity:

Here is a second diagram with A and B swapped:

Another way of looking at this second diagram is to think about necessity: the diagram shows us that A is necessary for B; that is to say, we must be inside the A circle in order to have any chance of being inside the B circle. What is important to note here though is that being inside the A circle is not enough [i.e. is not sufficient] by itself to guarantee we are also inside the B circle. So, we need A in order for B to be true, but A alone is not enough to guarantee B is true – that is what necessity is all about.

Again, reconcile this with your formal understanding of A is necessary for B, A ⟸ B

Finally, what happens if A is necessary and sufficient for B? If we look at our diagrams, we see that the A and the B circle need to be covering each other; in other words, the A circle and the B circle are the very same circle and that means they are really logically equivalent. So, A is necessary and sufficient for B is another way of saying A iff B or even A ⟺ B

Here is a summary of the notions of necessity and sufficiency:

A is sufficient for B	A only if B, if A then B
A is necessary for B	A if B, if B then A
A is necessary and sufficient for B	A iff B

A is sufficient for B	A ⟹ B
A is necessary for B	A ⟸ B
A is necessary and sufficient for B	A ⟺ B

Almost every TMUA paper will have some questions based on the terms necessary or sufficient. You should look through all the past papers and check you know how to deal with them and to check your understanding of the meaning of the two terms.

You should make sure you understand what each of the following terms means as they occur a lot in TMUA questions; use past paper questions and the model answers we have produced to help you:

necessary

sufficient

necessary and sufficient

necessary but not sufficient

sufficient but not necessary

not necessary and not sufficient [this is usually written as neither necessary nor sufficient, but we don’t write that in the TMUA as we want to be very clear]