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Definition 1.4.1
Intersection of sets A and B is a set A ∩ B
that consists of all elements that belong simultaneously to both sets A
and B, in symbolic notation
x ∈ A ∩ B iff x∈ A and x ∈ B.
Figure 1.4.1 Intersection of sets A and B.
Example 1.4.1
(a) Let A = {2i: i <16 and i ∈ N }, B = {3i: i<11 and i∈N}. Then A ∩ B consists of all natural numbers that are divisible by two and by three simultaneously, and are not greater than 30,
A ∩ B = {0, 6, 12, 18, 24, 30} = {6i : i<6, and ∈ N}.
(b) Let X be the set of all students of PJIIT, and Y be the set of all women. Then X ∩ Y is the set of all women-students.
By the definition, the intersection A ∩ B is a subset of both sets A and B. In particular it can be empty. In that case we say that the sets A and B are disjoint.
As a consequence of the definition 1.4.1, x∉ A ∩ B, when at least one of
its conditions is not fulfilled, in symbolic notation
x∉ A ∩ B iff x ∉ A or x ∉ B
Intersection is a binary operation that has properties of commutativity and associativity, and moreover distributivity with respect to intersection.
A ∩ (B ∩ C) = (A ∩ B) ∩ C (associative law)
A ∩ B = B ∩ A (commutative law)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (distributive law)
All the laws mentioned above remain analogous to addition and multiplication in the set of real numbers. There are however several other laws that are completely different.
A ∩ A = A (idempotent law)
A ∩ (A ∪ B) = A, A = A ∪ (A ∩ B) (absorption law)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (law of distribution of multiplication with respect to the union)
We shall illustrate the last law by the Venn's diagram .
Figure. 1.4.2 Illustration of the law of distribution.
As in the case of union, one can multiply inclusions. The following lemma presents this fact formally.
Lemma 1.4.1
If A ⊆ C and B ⊆ D, then A ∩ B ⊆ C ∩ D.
Proof. Let us assume that A ⊆ C and B ⊆ D. If x ∈ A ∩ B, then according to the definition of intersection, x ∈ A and x ∈ B. By the assumptions we have x ∈ C and x ∈ D, hence x ∈ C ∩ D. This implies that every element of the set A ∩ B belongs to the set C ∩ D, thus A ∩ B ⊆ C ∩ D. ♦
Question 1.4.2 Try to characterise inclusion by intersection.
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