College Algebra For First Year And Pre-Degree Students by T. G. Kulkarni, M. K. Kelkar

By T. G. Kulkarni, M. K. Kelkar

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I i i ) A is the set of multiples of 2 ( * > 0 if x e A) and C is the set of moltiples of 3 ( * > 0 if * e C ); h e n c e A U C is a set of multiples of 2, multiples of 3 and multiples of both 2 and 3 ( i. e. multiples of 6 ) ; i e. A JJ C = j 2, 4, 6, 8, .. a n d 3 , 9, 15, ... ( or A U C = j x | x is an even integer or an odd multiple of 3 ; * > 0 A f") C •* the set of common elements in A and C i. e. = { 6 , 1 2 , 18, Or j 4 f l C = ^ * | * i s a multiple of 6, * > 0 {. It may b e noted that both A U C and A (~) C are infinite sets.

If n ( A ) = r and n ( B) = s and A n B = 0, then the sum of two natural numbers r and s is given byr + J = n ( ^ U B ) . We shall now derive some fundamental properties of the natural numbers. ( i ) Closure Property. If disjoint sets A and B have r and s elements respectively then A U B has r + s elements; hence r + s is also a natural number. This shows that the addition of two n a t u r a l numbers is also a natural number. This fact is expressed by saying that the set of natural numbers is closed for REAL NUMBERS : 45 addition.

Hence 1 is the smallest natural number and it has no predecessor but it has a successor ; every other natural number has a unique ( i e. one and only o n e ) predecessor and a unique successer. If q, r, s, t e N. then we also have ( i i ) i f r > 5 , r + t>s + t, ( i i i ) if J">s, rt > st, ( i v ) i f r > s and tyq, r-\-t>s + q, ( v ) if r > 5 and tyq, rt>sq. These are known as order relations in N. 48 : COLLEGE ALGEBRA 2. The Number Corresponding to an Empty Set. , n = r. Thus, a natural number r can be assigned to every finite nonempty set by one-one correspondence with the set.

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