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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U 2 SETS : 2 9 ( i v ) We now consider here the operation of difference on two sets. If A and B are two sets then ( i ) A. A which do not belong to B and ( ii ) B — A is a set of all elements of B which do hot belong to A. Thus A - B = } *e>4; x e B ( and B - A = ) x\xe B-,x<£A\. (i) Shaded portion A — B. (ii j Shaded portion B — A . Fig. 12 Example. If A = ) 0, 1, 2 j and B = j 2, 3, 4 f, then A — B = set of elements which belong to to B. A - B = ) 0, M; and similarly B — A = J 3, 4 A but not Following identities on difference of sets may be noted.

22 (ii) SETS : 4 1 17. 18. ( I ) A U =A. (iv ) A 0 A' = 0, I. ii ) A n = 1>. I v * 0' = X, ( HI) A U -A = , (vi) (A'Y=A, (vii) A F|-X = A , (viii) A U X = X . ( i x ) A (1 A = A, ( * ) A L M ' = X, ( xi ) X' = ( x i i ) A — 0 = A> (xiii) A - A = 0 , (xiv) A f l B ' = A - (i ) B. B 1JC = ) 1, 2, 3, 8, 10 ••• A U < B U C ) = j 4, 5, 6. 7. 1, 2. 3, 8. 10 ( = ) 1, 2, 3, 4, 5, 6, 7, 8, 10 | ( il A U B = J 4, 5, 6. 7. 1. 2 , 3 ( A U - B ) U C = \ 4. 5. 6, 7, 1, 2, 3, 8. 10 [ = } 1. 2. 3. 4, 5, 6, 7 .

Hence unity i. e. 1 is called an identity element for multiplication. 1-3. Order in N. If we are given two different natural numbers r and s and it is known that there is another natural number t such that r + t - s, then we say that s is greater than r and write it as s > r or that r is less than s and write it as r < s. We can thus see that ( i ) for two different natural numbers r and s we have either r > 5 or r < 5. If A = ) 1, 2. , r then n ( A ) = r, B = ) 1,2, 3, , 5 then n ( B) = s, C = ) 1, 2, 3.

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