Relations and Functions

Relations and Functions

Question

Which of the following pairs of sets are disjoint?

(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }

(ii) { a, e, i, o, u } and { c, d, e, f }

(iii) {x : x is an even integer } and {x : x is an odd integer}

Answer

(i)  A = {1, 2, 3, 4},    
 
        space space space space space space space space space space space space space space space space space space space space space space space space space space space straight B space equals space space space space space space space space space left curly bracket space straight x thin space space colon space straight x space element of space straight N comma space space 4 space less or equal than space straight x space less or equal than space right curly bracket space equals space left curly bracket 4 comma space 5 comma space 6 right curly bracket

space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight A space intersection space straight B space equals space left curly bracket 1 comma space 2 comma space 3 comma space 4 right curly bracket space intersection space left curly bracket 4 comma space 5 comma space 6 right curly bracket space equals space left curly bracket 4 right curly bracket space equals straight ϕ
Hence, A and B are not disjoint sets.

(ii)               A = { a, e, i, o, u },   B =  { c, d, e, f }
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∴   A and B are not disjoint sets.

(iii)    A   =   {x : x is an even integer}
              = {........, -6, -4, -2, 0, 2, 4, 6, 8, 10, ................}
         B = {and {x : x is an odd integer} = {........., -5, -3, -1, 1, 3, 5, 7, .............}
space space space space space space space space space space rightwards double arrow space space space space space straight A space intersection space straight B space equals space left curly bracket space space space right curly bracket space equals space straight ϕ
Hence, A and B are disjoint sets. 

 

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