Operations on Sets · 4.3.2
Complement of combined set operations
Students first evaluate a combined set operation, such as A ∩ (B ∪ C), and then find its complement within the universal set ξ, every element not in that resulting set. This two-stage process tests both accurate combined-operation working and correct identification of the complement afterward.
The official learning standard (4.3.2)
“Determine the complement of combined operations on sets.”
What it means
Students first evaluate a combined set operation, such as A ∩ (B ∪ C), and then find its complement within the universal set ξ, every element not in that resulting set. This two-stage process tests both accurate combined-operation working and correct identification of the complement afterward.
How it is examined
This appears mostly in Paper 2 structured questions, where students calculate a combined operation and then state its complement as a final sub-part. Paper 1 may test a simpler version, asking students to identify the shaded complement region of a combined operation directly from a Venn diagram.
Worked example
The universal set is ξ = {1, 2, 3, ..., 10}. Set A = {1, 2, 3, 4, 5}, set B = {4, 5, 6, 7} and set C = {7, 8, 9}.
Find [A ∩ (B ∪ C)]′.
- Work out the bracket first: B ∪ C = {4, 5, 6, 7} ∪ {7, 8, 9} = {4, 5, 6, 7, 8, 9}.
- Find A ∩ (B ∪ C): compare A = {1, 2, 3, 4, 5} with {4, 5, 6, 7, 8, 9}; the common elements are 4 and 5, so A ∩ (B ∪ C) = {4, 5}.
- Find the complement in ξ: remove 4 and 5 from ξ = {1, 2, ..., 10}, leaving {1, 2, 3, 6, 7, 8, 9, 10}.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Should I find the complement before or after the combined operation?
Always after. Complete the entire combined operation, such as A ∩ (B ∪ C), to get one final resulting set first.
Only then find the complement of that whole result within ξ, taking the complement of an individual set too early will give a completely wrong final answer.
Is there a shortcut using De Morgan's laws instead?
De Morgan's laws, like (A ∩ B)′ = A′ ∪ B′, can simplify some expressions, but for SPM it's usually safer and clearer to compute the combined operation directly, then take one complement at the end, especially when working with listed elements rather than algebraic set expressions.
How can I check my answer for a complement of a combined operation?
Count the elements: the number of elements in the combined-operation set plus the number in its complement should equal n(ξ), the total number of elements in the universal set. If these two counts don't add up correctly, recheck your working for a missing or repeated element.