Question

For any sets A and B, prove that:
$A\cap B'=\phi \Rightarrow A\subset B$
$A'\cup B=U\Rightarrow A\subset B$

Answer 1

Hint: We will look at the Venn diagram for this question and we will also write some required definitions of the terms like union, intersection of sets and subset of a set.

Complete step-by-step answer:

Universal set: The set containing all objects or elements and of which all other sets are subsets.
Complement of a set: Complement of a set A, denoted by A c, is the set of all elements that belongs to the universal set but does not belong to set A.
Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .

Let’s start solving each part one by one.
For part (a):
From the above diagram and the definition we can see that $A\cap B'$ is a null set.
Therefore,
$A\cap B'=\phi $
And for this to be true we need $A\subset B$ as shown above.
Hence proved (a).
Let A = {1, 2}, B = {1, 2, 3}, U = {1, 2, 3, 4}
B’ = {4}
Now we will find,
$A\cap B'$ = $\phi $
Hence, by taking an example we have proved.
For part (b):
Similarly from the above diagram and definition we can see that $A'\cup B=U$
As it covers the whole universal set.
The part that is left by A’ is covered by B and for this to be true $A\subset B$must also be true.
Similarly we can take values of A, U and B and show that it is true.
Hence proved (b).
And hence both the statements have been proved.

Note: We have solved the above question by using the Venn diagram and some definitions. So, the use of a Venn diagram is very important and one should know how to draw it so that there is no mistake while solving.