Using the following please state which ones are used Make a

Using the following (please state which ones are used):

Make a conjecture and prove:

Definition 1 A set is a collection of objects. The otjects in a set X are called elements of X. Notation 2 A set can be described using set-builder notation. That is, a set can be described by uniting tac P(z) where P(r) is a statement concerning an olject r. For erample, N tz: z is a natural number) or (2 {r: is a real number and 2 r S Remark 3 Some authors use the notation (rlP(z) in place of tr P(z) Notation 4 The set that contains no elements is called the empty set and is denoted 2. /Note: is NOT the empty set Notation 5 If an o r is an element of a set A, then we write z E A. If r is not an bject element of A, then we write z A. Definition 6 Two sets A and B are equalif and only if every element of A belongs to Band every element of B belongs to A Definition 7 If A and Bare sets such that each element of A is also an element of B, then we say A is a subset of B, and notate this ASB. Remark 8 Some authors use the notation A CBif A is a subset of B. We reserve the notation C or something else later. Proposition 9 Let A and B be sets. Then A Bif and only if ACB and BCA. Definition 10 Let A be a set. The power set of A, denoted POA) or 2 is the set of all subsets of A.

Solution

!. set A has n element

as we know from above statement Set has 2 ⁿ    subsets

as we know in a power set A the no of element = the number of subset in set A = 2ⁿ

Let A be a set with n elements. the number of elements in the power set of A or P(A) is   2^n.

^{2. let we have 1 element in set A then the number of subset = 2 or =} 2¹

if Set a has k elements then number of subset is 2^k

Set A has k+1 element the then number of subset is euqal to number of subset of A has k element x number of subset of set A has 1 element= 2^{k x} 2= 2^k+1

^{henced proved}