Question

6. Prove that (A - B) UB= A iff BCA​

Answer 1

Answer:

$\boxed{\pink{\sf Hence , ( A - B ) \cup B = A }}$

Step-by-step explanation:

Given that , B is a subset of A . And we need to prove that , ( A - B ) U B = A . This can be done by taking some examples . We know that ,

• The term Subset means that here all the elements of Set B are in Set A . Let us take that , A = { 1 , 2 , 3 , 4 } . And B = { 1 , 2 } .

• The difference of two sets day A- B means that set of all those elements which are in A but not in B .

• The union of two sets say A and B is the set of all elements of sets A and B , the Common element being taken only once

• Here as per our assumption ,

$\sf\dashrightarrow A =\red{ \{ 1 , 2 , 3 , 4 \} }$

$\sf\dashrightarrow B =\red{ \{ 1 , 2 \} }$

Now by the definition of difference of two sets we have ,

$\sf\dashrightarrow A - B =\{ 1 , 2 , 3 , 4 \} - \{ 1 , 2\}$

• Now here the elements 3 and 4 are in A but not in B . Therefore ,

$\sf\dashrightarrow A - B =\red{\{ 3 , 4 \}}$

Now let's find out the union of A - B and B ,

$\sf\dashrightarrow ( A - B )\cup B = \{ 3, 4\} \cup \{ 1,2\} =\pink{\{ 1 ,2,3,4\}}$

And at RHS we have Set A which also equals to the above set . Hence Proved !