Question

For any two sets A and B prove that...
B is a subset of A union B. and
A union B is a subset of A.

Answer 1

Hey there!! the solution to your question is as follows:

Consider two sets namely 'A' and 'B'.
Let A contain some elements such that A = {1,2,3,4,5,6,7,8}
Similarly, let B contain some elements such that B = { 5,6,7,8}

then A U B ( A union B) = {1,2,3,4,5,6,7,8}
( In a union of two sets, we take all the distinct elements of two sets and form a new set that has the elements of both sets.

Now comparing set B with (A U B ), we find that all the elements of set B {5,6,7,8} are present in set (A U B). 
 
This makes set 'B' a subset of (A U B)
(When all the elements of a set are present in another set, then this set is called a subset of the other set.)

Now coming to the other part of the question. As we saw that Set 'A' in itself includes all the elements of (A U B) i.e {1,2,3,4,5,6,7,8} even before taking the union of set 'A' and set 'B'.

This means that all the elements of (A U B) are included in the set A. This makes (A U B), a subset of set A.

Hope it helps. :)

Answer 2

For two sets A and B such that A and B are overlapping and A⊂B, the given statements hold.

Given:

Two sets A and B.

To Find:

For any two sets A and B, we need to prove that:

i) B⊂ AUB

ii) AUB ⊂ A

Solution:

To solve this problem, we need to understand the following concepts:

1) A set is a collection of elements, separated by commas, and written inside curly brackets.

2) The union of two sets A and B gives us the elements that are contained in both sets.

3) A set A is said to be a subset of set B if every element of A is present in B.

4) Two sets are said to be disjoint if no element is common to both of them.

5) Two sets are said to be overlapping if there exists at least one element that is common to both of them.

We asked to prove, for any two sets A and B:

i) B⊂ AUB

ii) AUB ⊂ A

Let us prove the given statements by considering three possible cases of sets A and B:

Case 1: A = {1, 2, 3} and B = {4, 5, 6}

In this case, the sets are disjoint.

A U B = {1, 2, 3, 4, 5, 6}

We observe that B ⊂ AUB as every element of B is present in AUB.

Also, AUB ⊄ A as the elements 4, 5, and 6 are not present in set A.

Case 2: A = {1, 2, 3} and B = {2, 3}.

In this case, the sets are overlapping and B⊂ A.

A U B = {1, 2, 3, 4}

We observe that B ⊂ AUB as every element of B is present in AUB.

Also, AUB⊄A as element 4 is not present in set A.

Case 3: A = {1, 2, 3} and B = {1, 2, 3, 4}.

In this case, the sets are overlapping and A⊂B.

A U B = {1, 2, 3, 4}.

We observe that B⊂ AUB. Also, AUB⊂ A.

Hence, we conclude that both the given statements hold in case 3 where sets A and B are overlapping and A⊂B.

∴ For two sets A and B such that A and B are overlapping and A⊂B, the given statements hold.

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