Question

Deline set, subset. Union of two sets. Intersection of wo sets and cartesian product of two sets with example. Define equivalence relation, Partial and total order relation with example. Prove that set of real numbers is uncountable and union of countable sets is also countable ?​

Answer 1

Answer:

The union of two sets is a set containing all elements that are in A or in B (possibly both). For example, {1,2}∪{2,3}={1,2,3}. Thus, we can write x∈(A∪B) if and only if (x∈A) or (x∈B). Note that A∪B=B∪A. In Figure 1.4, the union of sets A and B is shown by the shaded area in the Venn diagram.

Union

Fig.1.4 - The shaded area shows the set B∪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.

Answer 2

Answer:

Define set subset.union of two sets.Intersection of two sets and cartesian product of two sets with example. Define equivalence relation.partial and total order relation with exampal.prove that set of real numbers is uncountable and union of countable sets is also countable.