If a > 0, prove that a(b,c) = (ab, ac). [One must assume that a > 0 lest a(b,c)
ID: 2975859 • Letter: I
Question
If a > 0, prove that a(b,c) = (ab, ac). [One must assume that a > 0 lest a(b,c) be negative.]Explanation / Answer
If a, b, and c are not all positive, then let's look at the possibilities: a) All three are negative, but then condition 1 will not match b) Only one of the numbers is negative, but then condition 3 will not match So the only thing left is for two of the numbers to be negative. So let's assume a and b are negative, then for condition1 to be true, we conclude that c > abs(a+b). Now looking at condition 2: ab + ac + bc > 0 ab + c(a + b) > 0 In order for this condition to be true, then ab (a positive number) must be greater than abs(c(a+b)), and since c is positive: c*abs(a+b). But we know that c > abs(a+b) therefore: c * abs(a + b) > abs(a + b) * abs(a + b) = a^2 + b^2 + 2ab > ab. A contradicton which will make cndition 2 false. Therefore the only possibility is for a, b and c to be positive.