SETS - Online Class 11 Maths NCERT Solutions

INTERSECTION AND UNION:

The set of elements common to two given sets A and B is known as the intersection and written as A ∩ B. The set of elements appearing in at least one of these sets is called the union, denoted by A ∪ B.

EXAMPLE: Decide which elements ought to belong to each of A ∪ B ∪ C and A ∩ B ∩ C. Then write a compact description of each set using bar notation.

Note that the set operation of intersection corresponds to the logical operation of conjunction. This relationship is made clear by the fact that

A ∩ B = {x | x ∈ A and x ∈ B}.

Similarly, union corresponds to the logical operation of disjunction, since

A ∪ B = {x | x ∈ A or x ∈ B}.

We say that A and B are equal sets, written A = B, if these two sets contain precisely the same elements. One common technique for showing that two sets are equal is to show that every element of the first set must be an element of the second set, and vice-versa.

To establish the set identity A ∩ B = A ∪ B we use these two strategies.

Step one: Let x be any element of the first set; i.e. let x ∈A ∩ B. This means that x “∈ A ∩ B. Since A ∩ B consists of elements in both A and B, if x is not in the intersection then either x “∈ A or x “∈ B, or both. In other words, x ∈ Aor x ∈ B, which means that x ∈ A ∪ B.

Step two: On the other hand, if x ∈ A ∪ B then we know that x ∈ A or x ∈ B, which means that x “∈ A or x “∈ B. Since x is missing from at least one of the sets A or B, it cannot reside in their intersection, hence x “∈ A∩B. Finally, this is the same as x ∈ A ∩ B. Hence, we conclude that the sets A ∩ B and A ∪ Bare indeed equal.

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