Answer :

Given:

A = {x ϵ W : x < 2}


Here, W denotes the set of whole numbers (non – negative integers).


A = {0, 1}


[ It is given that x < 2 and the whole numbers which are less than 2 are 0 & 1]


B = {x ϵ N : 1 < x ≤ 4}


Here, N denotes the set of natural numbers.


B = {2, 3, 4}


[ It is given that the value of x is greater than 1 and less than or equal to 4]


and C = {3, 5}


L. H. S = A × (B C)


By the definition of the union of two sets, (B C) = {2, 3, 4, 5}


= {0, 1} × {2, 3, 4, 5}


Now, by the definition of the Cartesian product,


Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e.


P × Q = {(p, q) : p Є P, q Є Q}


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


R. H. S = (A × B) (A × C)


Now, A × B = {0, 1} × {2, 3, 4}


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


and A × C = {0, 1} × {3, 5}


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


Now, we have to find (A × B) (A × C)


So, by the definition of the union of two sets,


(A × B) (A × C) = {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)}


= L. H. S


L. H. S = R. H. S is verified


(ii) Given:


A = {x ϵ W : x < 2}


Here, W denotes the set of whole numbers (non – negative integers).


A = {0, 1}


[ It is given that x < 2 and the whole numbers which are less than 2 are 0, 1]


B = {x ϵ N : 1 < x ≤ 4}


Here, N denotes the set of natural numbers.


B = {2, 3, 4}


[ It is given that the value of x is greater than 1 and less than or equal to 4]


and C = {3, 5}


L. H. S = A × (B C)


By the definition of the intersection of two sets, (B C) = {3}


= {0, 1} × {3}


Now, by the definition of the Cartesian product,


Given two non – empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, .i.e.


P × Q = {(p, q) : p Є P, q Є Q}


= {(0, 3), (1, 3)}


R. H. S = (A × B) (A × C)


Now, A × B = {0, 1} × {2, 3, 4}


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


and A × C = {0, 1} × {3, 5}


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


Now, we have to find (A × B) (A × C)


So, by the definition of the intersection of two sets,


(A × B) (A × C) = {(0, 3), (1, 3)}


= L. H. S


L. H. S = R. H. S is verified


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