Question

If $X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\} find
i.$X - Y$
ii.$Y - X$
iii.$X \cap Y$

Answer 1

To solve the given question where we need to find the subtraction between the two sets then we need to know how to determine the subtraction between them for example let us consider that A and B be two sets such that we need to find $A - B = A - \left( {A \cap B} \right)$, now we will simplify the above question using this formula.

Complete step-by-step answer:
We have given that
i.$X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\}
Now, using the relation discussed in the hint part are
$A - B = A - \left( {A \cap B} \right)$
Similarly
$X - Y = X - \left( {X \cap Y} \right)$
Let is first find the intersection between the sets that is
$X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\}
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\}
Intersection is the collection of the common elements in the sets
Hence, we get
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\} = \left\\{ {b,d} \right\\}
Now, subtracting from X we get
X - \left( {X \cap Y} \right) = \\{ a,b,c,d\\} - \left\\{ {b,d} \right\\} = \\{ a,c\\}
Hence the above represent the required result

ii.$X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\}
Now, using the relation discussed in the hint part are
$A - B = A - \left( {A \cap B} \right)$
Similarly
$Y - X = Y - \left( {Y \cap X} \right)$
Let is first find the intersection between the sets that is
$X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\}
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\}
Intersection is the collection of the common elements in the sets
Hence, we get
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\} = \left\\{ {b,d} \right\\}
Now, subtracting from X we get
Y - \left( {X \cap Y} \right) = \\{ f,b,g,d\\} - \left\\{ {b,d} \right\\} = \\{ f,g\\}
Hence the above represent the required result

iii.$X \cap Y$
Now , we need to find the common elements to find the intersection then in that case
Let is first find the intersection between the sets that is
$X = \\{ a,b,c,d\\}$and Y = \left\\{ {f,b,d,g} \right\\}
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\}
Intersection is the collection of the common elements in the sets
Hence, we get
X \cap Y = \\{ a,b,c,d\\} \cap \left\\{ {f,b,d,g} \right\\} = \left\\{ {b,d} \right\\}
Hence, the above represent the required result
Note: The intersection of two sets means the collection of the common element between them is known as intersection
For example
$A \cap B$
In the above we need to find the intersection that means we need to find the common elements in the set
A = \left\\{ {3,5,7,9,11} \right\\}
And
B = \left\\{ {7,9,11,13} \right\\}
If we have a look for the common elements that comes out to be $7,9,11$
Hence, the collection of the common elements between two sets is known as the intersection
A \cap B = \left\\{ {3,5,7,9,11} \right\\} \cap \left\\{ {7,9,11,13} \right\\} = \left\\{ {7,9,11} \right\\}
Hence, the above represent the required result.