a We want to determine whether each of the given inequalities connected with the diagram must be true. Remember that this way of presenting sets does not describe their sizes. This means that even though the circles on the diagram have the same size — it does not mean that the sets have the same number of elements.
We want to determine whether it must be true that H ≥ E. H is the set of students with brown hair.
E is the set of students with brown eyes.
We want to compare the sizes of these two sets. Since we have no idea how many students have brown hair or how many students have brown eyes, we cannot compare the sizes of H and E. Therefore, the given inequality is not necessarily true.
b Again we want to compare the sets of students with brown hair H with the set of students with brown eyes E. This time we consider an inequality H+10 ≥ E.
Remember that this way of presenting sets does not describe their sizes. This means that even after adding 10 students to set H we still cannot determine what is greater — the number of students in set H with additional 10 students or the number of students in set E. Therefore, inequality H+10 ≥ E does not have to be true.
c This time we want to determine whether inequality H ≥ X must be true. Notice that set X is a subset of H, which means that all elements of X are also elements of H.
Since all elements of X are also elements of H, the number of students in set X is less than or equal to the number of elements of set H. This means that inequality H ≥ X must be true.
d From Part C we already know that inequality H ≥ X must be true. If the size of H is greater than or equal to X, then adding 10 to H makes it even greater and it cannot create a false statement. Therefore H+10 is greater than or equal to X and the given inequality holds true.
e Now, we want to decide whether inequality H > X must be true. From Part C we know that set X is a subset of H, so H ≥ X. For the inequality to be strict, we need to know that set H is greater than set X, which means that there are students with brown hair but not with brown eyes.
Remember that this way of presenting sets does not describe their sizes. This means that we cannot be sure that there are any students with brown hair but not with brown eyes. Therefore, the given inequality is not necessarily true.
f From Part C we already know that inequality H ≥ X must be true. If the size of H is greater than or equal to X, then H+10 must be greater than X. This means that the number of students in set H with additional 10 students is strictly greater than the number of students in set X. Therefore the given inequality holds true.
