This problem is related to Chapter-Sets. Please solve the question using x method. x method means x ∈ A ∩ B
Q. For any sets A and B. Prove that A∩B⊂A and A∩B⊂B.
To prove that A∩B⊂A, we need to show that every element x in the intersection of A and B is also an element of A. Similarly, to prove A∩B⊂B, we need to show that every element x in the intersection of A and B is also an element of B.
The intersection of two sets A and B, denoted by A∩B, is defined as the set containing all elements that are common to both A and B. In symbols, A∩B = {x | x ∈ A and x ∈ B}.
Now, let's prove A∩B⊂A:
1. Take an arbitrary element x from A∩B. This means x ∈ A and x ∈ B, by the definition of the intersection.
2. Since x is an element of A, we can conclude that x ∈ A, which satisfies the hypothesis that every element x in A∩B is also an element of A.
Therefore, A∩B⊂A is proven.
Similarly, we can prove A∩B⊂B:
1. Take an arbitrary element x from A∩B. This means x ∈ A and x ∈ B, by the definition of the intersection.
2. Since x is an element of B, we can conclude that x ∈ B, which satisfies the hypothesis that every element x in A∩B is also an element of B.
Therefore, A∩B⊂B is proven.
To summarize, we have proven that for any sets A and B, A∩B⊂A and A∩B⊂B.