As we have already discussed that the union of two sets in mathematics is a set contains the elements of both sets and the intersection of two sets in mathematics is a set that contains common elements of both sets. Now, we are going to learn about the union and intersection of more than two sets in mathematics.
Union of three sets in mathematics:
The union of three sets can be obtained by following the steps given below:
First of all, find the union of any two sets.
Secondly, find the union of the remaining set with that set which is obtained in the first step.
Examples:
Properties implying for the union of sets in mathematics:
In mathematics, there are three types of properties Commutative Property, Associative Property, and Identity Property that are applied for the union of more than two sets that we are going to study now.
Commutative property with respect to the union of sets:
In commutative property, the operation of union is applied on two sets only. For example, If V and W are any two sets then the commutative property of union of two sets is applied as:
“VUW=WUV”
Let V = {-1, -3, -5} and W = {2, 4, 6} then we can prove this property as:
L.H.S = VUW
VUW = {-1, -3, -5} U {2, 4, 6}
= {-1, 2, -3,4 ,-5 ,6}.................(i) and
R.H.S = WUV
WUV = {2, 4, 6} U {-1, -3, -5}
= {-1, 2, -3, 4, -5, 6}................(ii)
By (i) and (ii)
Hence L.H.S = R.H.S here we have proved that commutative property for the union of two sets is verified for V and W.
Daily life example for the union of sets in mathematics:
Associative Property For the union of sets in mathematics:
In the associative property, the operation of union is applied on three sets. For Example, If D, E, and F are three sets then the Associative property for the union of three sets can be applied as:
“D U (E U F) = (D U E) U F”
For D = {a, e, f}, E = {r, s, t}, and F = {k, l, m} then it can be verified as:
L.H.S = DU(EUF) here, firstly we have to solve the bracket part.
(EUF) = {r, s, t} U {k, l, m}
= {k, l, m, r, s, t}
Now, we have to take the union of the above resultant part with the remaining part i.e,
DU(FUF) = {a, e, f} U {k, l, m, r, s, t}
= {a, e, f, k, l, m, r, s, t}...........................(i)
R.H.S = (DUE)UF
Similarly, (DUE) = {a, e,f} U {r, s, t}
= {a,e, f, r, s, t}
And, (DUE)UF = {a, e, f, r, s, t} U {k, l, m}
= {a, e, f, r, s, t, k, l, m}...................(ii)
By (i) and(ii)
Hence L.H.S = R.H.S we proved here that the Associative property for the union of two sets holds on D, E, and, F.
Example
Identity Property For the union of sets in mathematics
The word identity means sameness and this property for the union of sets is based on sameness. In mathematics, the empty set plays the role of identity for the union of sets. When we take the union of any set with an empty set then we obtain the same set in the result.
That is, for any set S the identity property for the union of sets can be applied as:
SU∅ = S
Now, let us take an example to prove this property.
L.H.S = SU∅
If S = {3, 6, 9} be any set then
SU∅ = {3, 6, 9} U {∅}
= {3, 6, 9}...............(i)
R.H.S= S
S= {3, 6, 9}.....................(ii)
By (i) and (ii)
L.H.S = R.H.S which shows that identity property holds for the union of sets in mathematics.
Daily Life examples of identity property for the union of sets:
Properties Implying to the Intersection of sets in mathematics:
The intersection of sets is a set that contains all the common elements of given sets. The intersection of three sets in mathematics can be found out by using the following steps:
Find out the intersection of any two sets.
Find out the intersection of the above resultant set with the remaining set.
Following properties are implied to the intersection of sets in mathematics that are:
Commutative Property for the intersection of sets:
The commutative property for intersection deals with two sets as we have studied for the union of sets. If C and D are any two sets then the commutative property for the intersection of sets states that:
“C ⋂ D = D ⋂ C”
For Example If C = {2, 3, 5, 7} and D = {2, 4, 6} then commutative property can be applied as:
L.H.S = C ⋂ D
C ⋂ D = {2, 3, 5, 7} ⋂ {2, 4, 6}
= {2}...............(i)
R.H.S = D ⋂ C
D ⋂ C = {2, 4, 6} ⋂ {2, 3, 5, 7}
={2}................(ii)
By (i) and (ii)
Hence L.H.S = R.H.S which shows that commutative property for the intersection of sets holds for the sets C and D.
Examples
Associative Property for the Intersection of sets:
The associative property for the intersection of three sets R, S, and T can be applied as:
“R ⋂ (S ⋂ T) = (R ⋂ S) ⋂T”
For R = {l, m, n}, S= {p, l, m}, T= {m, n, o} then
L.H.S = R ⋂ (S ⋂ T)
(S ⋂ T) = {p, l, m} ⋂ {m, n, o}
= {m}
R ⋂ (S ⋂ T) = {l, m, n} ⋂ {m}
= {m}..............................(i)
R.H.S = (R ⋂ S) ⋂ T
(R ⋂ S) = {l, m, n} ⋂ {p, l, m}
= {l, m}
(R ⋂ S) ⋂ T = {l, m} ⋂ {m, n, o}
= {m}.............................(ii)
By (i) and (ii)
Hence L.H.S = R.H.S which shows that associative property for the intersection of sets R, S, and T is verified.
Examples
Identity Property For the Intersection of sets:
As we have studied above that empty set plays the role of identity in sets. But in the intersection of sets, this property can be applied as:
For a set G identity property states that
“G ⋂ ∅ =∅ ”
Let G = {4, 8, 12} then
L.H.S = G ⋂ ∅
G ⋂ ∅ = {4, 8, 12} ⋂ {∅}
= {∅}...................(i)
R.H.S = ∅
= {∅}.....................(ii)
By (i) and (ii)
Hence L.H.S = R.H.S which shows that identity property is proved for a set G.
Examples
Properties for Difference of sets in mathematics:
The difference between the two sets is a set that contains the elements of the first set that are not common between the two sets.
The property for the difference of two sets can be applied as:
For any two unequal sets U and V
“U - V ≠ V - U”
Suppose that U = {r, s, t} and V = {p, q r}
L.H.S = U - V
U - V = {r, s, t} - {p, q r}
= {s, t}...................(i)
V - U = {p, q r} - {r, s, t}
= {p, q}.................(ii)
By (i) and (ii)
Hence L.H.S ≠ R.H.S and the property for the difference of sets U and V is proved.
Examples
Practice Examples for the union and intersection of sets in mathematics
Question No. 1:
If F= {a,b, c, o, u}, G = {b, c, d} and H = {a, e, f, g}, then demonstrate the following statements are proved or not?
(i) G ∩ (H ∩ F) = (G ∩ H) ∩ F
(ii) G U (H U F) = (G U F) U F
(iii) F ∩ (G ∩ H) = (F ∩ G) ∩ H
(iv) F U (G U H) = (F U G) U H
(v) H ∩ (F ∩ G) = (H ∩ F) ∩ G
(vi) H U (F U G) = (H U F) U G
Question No.2
For K = {2, –1, 0, 1}, L= {–4, –2, 1, 3} and M= {0, ±2, ±4}, then tell whether the following statements are verified or not?
(i) K ∩ (L ∩ M) = (K ∩ L) ∩ M
(ii) K U (L U M) = (K U L) U M
(iii) L ∩ (K ∩ M) = (L ∩ K) ∩ M
(iv) L U (K U M) = (L U K) U M
(v) M ∩ (L ∩ K) = (M ∩ L) ∩ K
(vi) M U (L U K) = (M U L) U K
Question No.3
Let Q = {1, 7, 9, 11}, R = {1, 5, 9, 13}, and S = {2, 6, 9, 11}, then verify that: (i) Q - R ≠ R - Q
(ii) R - S ≠ S - R
(iii)Q U ∅ = Q
(iv)R U ∅ = R
(v) S U ∅ = S
(vi)Q ⋂ ∅ = Q
(vii)R ⋂ ∅ = R
(viii)S ⋂ ∅ = S