Lesson 9 of 11 · 7 min
Difference of sets
NCERT §1.9.3
The cricket coach wants the students free for extra practice: on the cricket list, but not on the football list.
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In short
The difference A − B (read "A minus B") = {x : x ∈ A and x ∉ B} holds the elements of A that are not in B.
Order matters: with A = {1, 2, 3, 4} and B = {3, 4, 5}, A − B = {1, 2} but B − A = {5}. In general A − B ≠ B − A.
The three pieces A − B, A ∩ B and B − A never share an element, and joined together they give exactly A ∪ B.
Any set splits along another: A is the part A − B together with the part A ∩ B, and attaching B − A to A gives the whole of A ∪ B.
A − B = φ exactly when A ⊂ B; and A − B = A exactly when A and B are disjoint.