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

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Extra info for College Algebra For First Year And Pre-Degree Students

Example text

This is indicated by crossed area 34 : COLLEGE ALGEBRA ( A' n B' y = A u B, indicated by two circles taken together. ( A ' U S ' ) ' is indicated by plain area i . e . common portion of two circles. («) (v) Rectangle indicates universal s e t . A - B =) x e A, x f c B , j plain area ( A — B )' = shaded area. («) (vi ) am Fig. 17 Rectangle indicates universal set. B - A = ) xe B; x £ A shaded portion. (B - A )' = plain area i. e. whole unshaded portion. Fig. 18 (6) (>) A is indicated by right slant lines.

For r, 5, t, e J the order relation satisfies the following properties : ( i ) if r < s, then r + t < s + t. ( ii ) If r < s and 0 < t, then r X t < s x ( i i i ) If r < s, and t = 0, then r X t — s X t. ( i v ) If r < 8 and t < 0, then r x t > s x t. 4. Rational Numbers. If r, s, t e J and r # 0 and r x s =t, then we define the number s as the quotient of t and r and write it as s = -y or s — t + r. This operation of finding the quotient is called division. Just as subtraction is the inverse operation of addition, we may consider division as the inverse operation of that of multiplication.

E. x s A fl A. AC Ad ( i ) A. Hence from ( i ) and ( i i ) A fl A = A. ( ii ) AC\ A' =. Proof: xsAHA'^xsAandxeA'. But A and A' are compliments of each other and have no common element and hence this set is A PI A' = . ( i i i ) Ad = (t>. P r o o f : xeAD0=>xsA and x e . But