Set Test - 2

If A−B=ϕ then this means all the elements of A are in B.

Similarly, If B−A=ϕ then this means all the elements of B are in A

Hence this implies A = B

Not all infinite sets are equivalent to each other.

For example: the set of all real numbers and the set of integers.

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

In some textbooks and popularizations, the empty set is referred to as the "null set". However null set is a distinct notion within the context of measure theory. In that setting, it describes a set of measure zero; such a set is not necessarily empty. The empty set may also be called the void set. Common notations for the empty set include "{}", "Φ", and "∅".

Before we define subset, we need to refresh ourselves on what a set is. A set is a collection of elements or objects. These elements are usually related in some way, but this is not necessary. However, each element needs to be discernible from the other elements of the set

So in set (D) all the elements of set A.

Given, order: P, Q, R, S, T, U, V.

The point is  A person can see caps of all persons standing in front of him but cant see his own cap and caps of persons standing behind him. S is able to see in front of him green and blue, but not violet which means the people in front of S i.e., P or Q or R worn green or blue and people after S and including himself i.e., S or T or U or V worn violet. T can see violet and yellow but not red.

Clearly, S is wearing violet cap and P/Q/R worn green/blue/yellow not necessarily in that order. V can see caps of all colours other than orange which means he is the last person in the row, he can see all caps except his. So, we can say V is wearing Orange cap. Given that T wears indigo cap.

The summary is:

Finally we left with red cap which means U is wearing it.

A set is said to be an infinite set whose elements cannot be listed if it has an unlimited (i.e. uncountable) by the natural number 1, 2, 3, 4, ………… n, for any natural number n is called a infinite set. A set which is not finite is called an infinite set.
 

let the elements of A = n 

and given that element of A is also element of B so n belongs to B also

which implies A is subset of B.

If set A represents all the cricket players of India and set B represents the cricket players who belong to the state of Maharashtra, then A ∩ B indicates all the cricket players other than the ones who belong to the state of Maharashtra.

When we have two sets that have the exact same elements, we call them equal sets. It does not matter what order the elements are in. It just matters that the same elements are in each set. So if the  power sets of them is same the sets are Equivalent. Hence if,

n[P(A)]=n[P(B)]

Then the sets are equal.

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written as {2, 4, 6}, which could also be written as {2, 6, 4}.

The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.

Glve: A={x∣x+4=4}

⇒A={x∣x=4−4=0}

⇒A={0} and B={0}

⇒A=B