Question

Which of the following pairs of sets are disjoint
i.\left\\{ {1,2,3,4} \right\\}\,and\,\left\\{ {x:x\,is\,a\,natural\,number\,and\,4 \leqslant x \leqslant 6} \right\\}
ii.\left\\{ {a,e,i,o,u} \right\\}\,and\,\left\\{ {c,d,e,f} \right\\}
iii.\left\\{ {x:x\,is\,an\,even\,\operatorname{int} eger} \right\\}\,and\,\left\\{ {x:x\,is\,an\,odd\,\operatorname{int} eger} \right\\}

Answer 1

To find which of the above options are disjoint then in that case we need to recall the word intersection between sets which means collection of common elements in the sets are called intersection and in disjoint if the intersection shows no common elements.

Complete step-by-step answer:
Let us solve the solution by finding the intersection step by step
i.\left\\{ {1,2,3,4} \right\\}\,and\,\left\\{ {x:x\,is\,a\,natural\,number\,and\,4 \leqslant x \leqslant 6} \right\\}
Let is consider the above two set and we need to check if the above two sets are disjoint or not then in that case we will first find the intersection between them
\,\left\\{ {x:x\,is\,a\,natural\,number\,and\,4 \leqslant x \leqslant 6} \right\\} = \left\\{ {4,5,6} \right\\}
And now we need to find the intersection between the sets
Let us consider that
A = \left\\{ {1,2,3,4} \right\\}
And
B = \left\\{ {4,5,6} \right\\}
And the intersection between them are
A \cap B = \left\\{ {1,2,3,4} \right\\} \cap \left\\{ {4,5,6} \right\\} = \left\\{ 4 \right\\}
Hence the intersection have a single common element
So, we can say that the given two set are not disjoint

ii.\left\\{ {a,e,i,o,u} \right\\}\,and\,\left\\{ {c,d,e,f} \right\\}
Let us consider that
A = \left\\{ {a,e,i,o,u} \right\\}
And
B = \left\\{ {c,d,e,f} \right\\}
Let is consider the above two set and we need to check if the above two sets are disjoint or not then in that case we will first find the intersection between them
A \cap B = \left\\{ {a,e,i,o,u} \right\\} \cap \left\\{ {c,d,e,f} \right\\} = e
The above shows that the two sets have a common element hence it is not disjoint

iii.\left\\{ {x:x\,is\,an\,even\,\operatorname{int} eger} \right\\}\,and\,\left\\{ {x:x\,is\,an\,odd\,\operatorname{int} eger} \right\\}
Let us consider that
A = \left\\{ {x:x\,is\,an\,even\,\operatorname{int} eger} \right\\}
And
B = \left\\{ {x:x\,is\,an\,odd\,\operatorname{int} eger} \right\\}
As we know that the integers are the number which have the elements which is even as well as the element in them is odd hence, if we have a look then the odd numbers will not present in even integers and vice versa then in that case the result is the above set is disjoint.

Note: In the above question the set of integers lies between the negative value to the positive value that means the odd numbers will be $- 3, - 1,1,3,5$ and so on and taking a eye on even integers we have $- 4, - 2,2,4,6$ and so on that makes it clear that they are disjoint set because they don’t have any common elements in them.