Question

State whether each of the following statement is true or false
i.\[\{ 2,3,4,5\} \,and\, \{ 3,6\} \,\]are disjoint sets
ii.\[\{ a,e,i,o,u\} \,and \{ a,b,c,d\} \] are disjoint sets
iii.\[\{ 2,6,10,14\} \,and \{ 3,7,11,15\} \] are disjoint sets
iv.\[\{ 2,6,10\} \,and\, \{ 3,7,11\} \] are disjoint sets

Answer 1

Hint: To solve the given type of question which involves the information of disjoint set then in that case the disjoint definition is used to solve such type of question which is the sets are known to be disjoint if they the intersection between the sets involves no common elements in them that means we need to find the intersection between them.

Complete step-by-step answer:
To solve the disjoint set
i.\[\{ 2,3,4,5\} \,and\, \{ 3,6\} \,\]are disjoint set then in that case let us find the intersection between two sets
Let us consider
\[A = \{ 2,3,4,5\} \]
And
\[B = \,\{ 3,6\} \,\]
Intersection is known as the collection of common elements between the set so, to find the collection we need to find the common elements of the above sets
$A \cap B = \{ 2,3,4,5\} \cap \{ 3,6\} = \{ 3\} $
That means the set is not disjoint
Hence, the above statement is false

ii.\[\{ a,e,i,o,u\} \,and\{ a,b,c,d\} \]are disjoint set then in that case let us find the intersection between two sets
Let us consider
\[A = \{ a,e,i,o,u\} \,\]
And
\[B = \,\{ a,b,c,d\} \]
Intersection is known as the collection of common elements between the set so, to find the collection we need to find the common elements of the above sets
$A \cap B = \{ a,e,i,o,u\} \, \cap \{ a,b,c,d\} = \{ a\} $
That means the set is not disjoint
Hence, the above statement is false

iii.\[\{ 2,6,10,14\} \,and \{ 3,7,11,15\} \] Are disjoint sets
Let us consider
\[A = \{ 2,6,10,14\} \,\]
And
\[B = \,\{ 3,7,11,15\} \]
Intersection is known as the collection of common elements between the set so, to find the collection we need to find the common elements of the above sets
$A \cap B = \{ 2,6,10,14\} \, \cap \{ 3,7,11,15\} = \phi $
That means the set are disjoint
Hence, the above statement is true

iv.\[\{ 2,6,10,14\} \,and \{ 3,7,11,15\} \] Are disjoint sets
\[\{ 2,6,10\} \,and\, \{ 3,7,11\} \]are disjoint sets
Let us consider
\[A = \{ 2,6,10\} \,\]
And
\[B = \{ 3,7,11\} \]
Intersection is known as the collection of common elements between the set so, to find the collection we need to find the common elements of the above sets
$A \cap B = \{ 2,6,10\} \, \cap \,\{ 3,7,11\} = \phi $
That means the set are disjoint
Hence, the above statement is true.

Note: In the given type of question where the intersection does not have any common elements then in that case the intersection is denoted by phi that represents the vacant sets such that it does not have any elements in it.