Answer

Verified

446.1k+ views

**Hint:**In order to solve such a problem, we must know that what $A \times B$ means and symbolises.

Here this refers to the Cartesian cross product and this includes all the sets of the ordered pairs. This will be clearer by an example. So let us take one example: let us suppose we have the elements in the sets $A,B$ where $A = \{ a,b,c\} $ and $B = \{ d,e\} $

So $A \times B = \{ (a,d),(a,e),(b,d),(b,e),(c,d),(c,e)\} $

And we must also know that the number of elements in the Cartesian product is simply equal to the multiplication of the number of elements in the set$A$ with the number of elements in the set$B$.

$n(A \times B) = n(A) \times n(B)$

And we must know the symbol $A \cup B$ which includes all the elements of $A,B$ but if the same element is in both the sets then it is taken only once.

**Complete step-by-step answer:**

Here we are given in the question the three sets which are

$A = \{ x \in W|x < 2\} $

$B = \{ x \in N|1 < x \leqslant 4\} $

$C = \{ 3,5\} $

So we can get the elements of these sets as we are given that $A$ contains the whole number which are less than $2$

So

$A = \{ 0.1\} $

Set $B$contains the natural number which are greater than one but less than or equal to $4$

$B = \{ 2,3,4\} $

$C = \{ 3,5\} $

So by the definition of the Cartesian product we get that

$B \cup C = \{ 2,3,4,5\} $

$A \times [B \cup C]$$ = \{ 0,1\} \times \{ 2,3,4,5\} $

$ = \{ (0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)\} $

So solving for $[A \times B] \cup [A \times C]$

$[A \times B]$$ = \{ 0,1\} \times \{ 2,3,4\} $

$ = \{ (0,2),(0,3),(0,4),(1,2),(1,3),(1,4)\} $

$[A \times C]$$ = \{ 0,1\} \times \{ 3,5\} $

$ = \{ (0,3),(0,5),(1,3),(1,5)\} $

$[A \times B] \cup [A \times C]$

$ = \{ (0,2),(0,3),(0,4),(1,2),(1,3),(1,4)\} \times $$\{ (0,3),(0,5),(1,3),(1,5)\} $

The terms which are in both the sets are to be taken only one time according to the definition of union of the two sets.

$[A \times B] \cup [A \times C]$$ = \{ (0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(0,5),(1,5)\} $

$ = \{ (0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)\} $

$ = A \times [B \cup C]$

**Hence we proved that $A \times [B \cup C]$$ = [A \times B] \cup [A \times C]$**

**Note:**Here in these type of questions we must take care that if the two terms are same in both the sets and we take the union of them then we need to take that term only once otherwise the answer may be incorrect and we must know what the following terms signifies in the sets like

$

A \cup B \\

A \cap B \\

A - B \\

B - A \\

$

Because without the knowledge of these terms we will be unable to solve such type of problems in sets.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

What percentage of the solar systems mass is found class 8 physics CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE