. If U=A={1,3,5,7} , B= {1,2,3,6} and C={1,2,4,6} then prove the Distributive law of union over intersection.
Answers
Answer:
Fig.1.4 - The shaded area shows the set $B \cup A$.
Similarly we can define the union of three or more sets. In particular, if A1,A2,A3,⋯,An are n sets, their union A1∪A2∪A3⋯∪An is a set containing all elements that are in at least one of the sets. We can write this union more compactly by
⋃i=1nAi.
For example, if A1={a,b,c},A2={c,h},A3={a,d}, then ⋃iAi=A1∪A2∪A3={a,b,c,h,d}. We can similarly define the union of infinitely many sets A1∪A2∪A3∪⋯.
The intersection of two sets A and B, denoted by A∩B, consists of all elements that are both in A and−−− B. For example, {1,2}∩{2,3}={2}. In Figure 1.5, the intersection of sets A and B is shown by the shaded area using a Venn diagram.
Intersection
Fig.1.5 - The shaded area shows the set B∩A.
More generally, for sets A1,A2,A3,⋯, their intersection ⋂iAi is defined as the set consisting of the elements that are in all Ai's. Figure 1.6 shows the intersection of three sets.
Intersection of 3 sets
Fig.1.6 - The shaded area shows the set A∩B∩C.
The complement of a set A, denoted by Ac or A¯, is the set of all elements that are in the universal set S but are not in A. In Figure 1.7, A¯ is shown by the shaded area using a Venn diagram.
Complement
Fig.1.7 - The shaded area shows the set A¯=Ac.
The difference (subtraction) is defined as follows. The set A−B consists of elements that are in A but not in B. For example if A={1,2,3} and B={3,5}, then A−B={1,2}. In Figure 1.8, A−B is shown by the shaded area using a Venn diagram. Note that A−B=A∩Bc.
Difference
Fig.1.8 - The shaded area shows the set A−B.
Two sets A and B are mutually exclusive or disjoint if they do not have any shared elements; i.e., their intersection is the empty set, A∩B=∅. More generally, several sets are called disjoint if they are pairwise disjoint, i.e., no two of them share a common elements. Figure 1.9 shows three disjoint sets.
Disjoint sets
Fig.1.9 - Sets A,B, and C are disjoint.
If the earth's surface is our sample space, we might want to partition it to the different continents. Similarly, a country can be partitioned to different provinces. In general, a collection of nonempty sets A1,A2,⋯ is a partition of a set A if they are disjoint and their union is A. In Figure 1.10, the sets A1,A2,A3 and A4 form a partition of the universal set S.
Partition
Fig.1.10 - The collection of sets A1,A2,A3 and A4 is a partition of S.
Here are some rules that are often useful when working with sets. We will see examples of their usage shortly.
Theorem : De Morgan's law
For any sets A1, A2, ⋯, An, we have
(A1∪A2∪A3∪⋯An)c=Ac1∩Ac2∩Ac3⋯∩Acn;
(A1∩A2∩A3∩⋯An)c=Ac1∪Ac2∪Ac3⋯∪Acn.
Theorem : Distributive law
For any sets A, B, and C we have
A∩(B∪C)=(A∩B)∪(A∩C);
A∪(B∩C)=(A∪B)∩(A∪C).
Example
If the universal set is given by S={1,2,3,4,5,6}, and A={1,2}, B={2,4,5},C={1,5,6} are three sets, find the following sets:
A∪B
A∩B
A¯¯¯¯
B¯¯¯¯
Check De Morgan's law by finding (A∪B)c and Ac∩Bc.
Check the distributive law by finding A∩(B∪C) and (A∩B)∪(A∩C).
Solution
A Cartesian product of two sets A and B, written as A×B, is the set containing ordered pairs from A and B. That is, if C=A×B, then each element of C is of the form (x,y), where x∈A and y∈B:
A×B={(x,y)|x∈A and y∈B}.
For example, if A={1,2,3} and B={H,T}, then
A×B={(1,H),(1,T),(2,H),(2,T),(3,H),(3,T)}.
Note that here the pairs are ordered, so for example, (1,H)≠(H,1). Thus A×B is not the same as B×A.
If you have two finite sets A and B, where A has M elements and B has N elements, then A×B has M×N elements. This rule is called the multiplication principle and is very useful in counting the numbers of elements in sets. The number of elements in a set is denoted by |A|, so here we write |A|=M,|B|=N, and |A×B|=MN. In the above example, |A|=3,|B|=2, thus |A×B|=3×2=6. We can similarly define the Cartesian product of n sets A1,A2,⋯,An as
A1×A2×A3×⋯×An={(x1,x2,⋯,xn)|x1∈A1 and x2∈A2 and ⋯xn∈An}.
The multiplication principle states that for finite sets A1,A2,⋯,An, if
|A1|=M1,|A2|=M2,⋯,|An|=Mn,
then
∣A1×A2×A3×⋯×An∣=M1×M2×M3×⋯×Mn.
An important example of sets obtained using a Cartesian product is $\mathbb{R}^n$, where $n$ is a natural number. For $n=2$, we have
$\mathbb{R}^2$ $= \mathbb{R}\times \mathbb{R}$
$= \{(x,y) | x \in \mathbb{R}, y \in \mathbb{R} \}$.
Thus, $\mathbb{R}^2$ is the set consisting of all points in the two-dimensional plane. Similarly, $\mathbb{R}^3=\mathbb{R}\times \mathbb{R} \times \mathbb{R}$ and so on.
Answer:
plz tell me if u get answer from miss