Tiêu đề 04_VI_E-TEC_VOL-2_SET THEORY_PG_134-174.pdf ✅

e-Techno Text Book VI-Mathematics (Vol- 2) 135 Observe the following example: A circle is defined as a set of all those points which move in such a way that their distance from a fixed point always remains constant. Here, we represent a circle with centre ‘O’ and radius ‘r’ as { P : OP= r , where ‘O’ is a fixed point}. Equal sets and its example: Two sets ‘A’ and ‘B’ are said to be equal, if every element of ‘A’ is in B and every element of B is in A and we write A = B. Example: Let A = set of letters in the word, wolf and B = set of letters in the word, follow. i.e., A = { w, O, l, f } and B = { f, O, l, l, 0, w } = { f, O, l, w} In the set B, the elements l, O are repeated elements. We do not consider repeti- tion of elements while writing the elements of a set. Clearly, A = B, since every element of ‘A’ is in B and every element of ‘B’ is in A. Cardinal number of a set and its examples: The number of distinct elements in a set ‘A’ is called the cardinal number of A, de- noted by n(A). Example: B = set of letters in the word ‘mathematics’. Then B = {m, a, t, h, e, i, c, s} . So, n(B) = 8. Equivalent sets: Two sets ‘A’ and ‘B’ are said to be equivalent if n(A) = n(B) and we write A  B. Let A = set of all odd numbers less than 10 and B = {2, 4, 6, 8, 10} i.e., A = {1, 3, 5, 7, 9} = i.e., n(A) = 5 and B = {2, 4, 6, 8, 10} i.e., n(B) = 5  n(A) = n(B) = 5  A  B Empty set or null set: A set having no element at all is called an empty set or null set. It is denoted by  . Eg: A = {x/x is a natural number, 1 < x < 2} Here, the elements greater than 1 cannot be less than 2  There is no such element which satisfies the given condition. A =  . Non - empty set: A set containing atleast one element is called a non-empty set. Eg: A = {x/x is a composite number, less than 10} i.e., A = { 4, 6, 8, 9} i.e., A    ‘A’ is a non-empty set. Singleton set: A set containing exactly one element is called a singleton set. Thus ‘A’ is a single- ton set, if n(A) = 1. Eg: The set of even prime numbers i.e., {2}, is a singleton set. Finite set: A set having ‘no’ element or a definite number of elements is called a finite set.