Sets
ICSE Class 8 · Mathematics · Chapter 6
What this chapter covers
- Sets: a quick recap
- The universal set
- Venn diagrams
- Union of sets
- Intersection of sets
- Disjoint sets
- Complement of a set
- Putting the operations together with Venn diagrams
Key points
- A set is a well-defined collection of distinct objects, written in roster form or set-builder form.
- The cardinal number n(A) counts the distinct elements of a finite set; n(emptyset) = 0.
- The universal set U contains every element under discussion and is chosen for each problem.
- A Venn diagram shows U as a rectangle and each set as a closed shape inside it.
- A cup B holds the elements in A, in B, or in both; each common element is written once.
- A cap B holds only the elements common to A and B.
- Disjoint sets have no common element, so A cap B = emptyset and n(A cup B) = n(A) + n(B).
- A' holds the elements of U not in A, and n(A') = n(U) - n(A).
- A cap A' = emptyset, A cup A' = U, (A')' = A, U' = emptyset and emptyset' = U.
- In a two-set Venn diagram, every element of U lies in exactly one of four regions.
Questions answered
Is "the collection of interesting films" a set? Give a reason.
No. Whether a film is interesting depends on who is judging, so the collection is not well-defined.
If A = \1, 3, 5, 7\ and B = \3, 7, 9\, find A cup B and A cap B.
A cup B = \1, 3, 5, 7, 9\ and A cap B = \3, 7\.
Are the sets \x : x is a multiple of 5\ and \x : x is a multiple of 7\ disjoint? Explain.
No. The number 35 is a multiple of both 5 and 7, so it belongs to both sets and their intersection is not empty.
Let U = \a, b, c, d, e, f, g\ and P = \a, c, e, g\. Find P' and n(P').
P' = \b, d, f\ and n(P') = 3.
Write down A cap A' and A cup A' for any set A in a universal set U.
A cap A' = emptyset and A cup A' = U.
Two sets M and N are disjoint, with n(M) = 9 and n(N) = 5. What is n(M cup N)?
Because the sets share no element, n(M cup N) = 9 + 5 = 14.
In a Venn diagram, which region shows A cap B, and which region shows (A cup B)'?
A cap B is the overlap of the two circles. (A cup B)' is the part of the rectangle outside both circles.
Let U = \1, 2, 3, ldots, 10\, A = \2, 4, 6, 8, 10\ and B = \1, 2, 3, 4, 5\. Find A' cap B.
A' = \1, 3, 5, 7, 9\, so A' cap B = \1, 3, 5\.