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SM 1. Expand and compute the following double sums mr rs 2. Consider a group of individuals each having a certain number of units of m different goods. Let , m; j-1,. ,n) aij denote the number of units of good i owned by person j i Explain in words the meaning of the following sums nm (a) Σaij (b) Xaj SECTION 3.4 A FEW ASPECTS OF LOGIC 61 3. Prove that the sum of all the numbers in the triangular table a1i a21 a22 d31 a32 a33 m am2 am3amm Σ(Σ,,) and also as ΣΣ@j) can be written as HARDER PROBLEM SM 4. Consider the m n numbers a in the rectangular array (I). Denote the arithmetic mean of them all by a, and the mean of all the numbers in the jth column by aj, so that FIE mn Prove that a is the mean of the column sums āj (j = l, . . . , n) and that 12

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