Subject: Elementary Mathematics
Q.1 Define difference and symmetric difference of two sets and prove that .
(A− B)U(B − A) = (AUB)−(A∩ B)
Difference of Two Sets :
The difference of the sets A and B in this order is the set of elements which belong to A but not to B. Symbolically, we write A – B and read as “ A minus B”.
The representation of A – B using a Venn diagram is given below.
Similarly, we can find B – A, the difference of the sets B and A in this order is the set of elements which belong to B but not to A. Symbolically, we write B – A and read as “ B minus A”. and a Venn diagram for B – A as:
Also, note that A – B is not equal to B – A, i.e. A – B ≠ B – A.
Symmetric Difference between Two Sets:
The set which contains the elements which are either in set A or in set B but not in both is called the symmetric difference between two given sets. It is represented by A ⊝ B and is read as a symmetric difference of set A and B.
We have to prove that : (A-B) U (B-A) = (A∪B) – (A∩B)
Proof:
Let, x ∈ (A-B) U (B-A)
⇒ x ∈ (A-B) or x ∈ (B-A)
⇒ x ∈ A But x ∉ B or x ∈ B but x ∉ A,
⇒ x ∈ A or x ∈ B
⇒ x ∈ (A∪B)
⇒ x ∈ (A∪B) – (A∩B)
Since here x represents the arbitrary element of the set (A-B) U (B-A).
Thus, (A-B) U (B-A) = (A∪B) – (A∩B)
Solution:
An equivalence relation is one which is reflexive, symmetric and transitive. R is an equivalence relation on set A. Let the element of set A be
So, – since it is reflexive, this is also true for It is symmetric hence, and this is also true for Also, R is transitive i.e., and For and or Thus is symmetric, reflexive and transitive. i.e. is equivalence Relation.Q.3 Find out the square root of complex numbers 12+5i
Solution :
√12–5i)=a+ib,
(a+ib)2=12–5i,
(a2−b2)+2abi=12–5i,
a2−b2=12, 2ab=−5,
(a2+b2)2=(a2−b2)2+(2ab)2
=122+52=144+25=169=132,
a2+b2=13,
2a2=12+13=25, 2b2=13−12=1,
a=±52–√, b=∓12–√, (ab<0)
a+ib=±12–√(5−i).