consider the following three possible conditions on two real numbers x and y

$p : x$ and $y$ satisfy the equation $(x+y)^2 = a(x^2+y^2) + bxy$ where $a,b$ are real constants.

$q : x = 0$ and $y = 0$

$r : x = 0$ or $y=0$

  1. suppose that in condition $p, a = b = 1$ then $p$ is [H] for $q$ and $p$ is [I] for $r$.
  2. suppose that in condition $p, a = b = 2$ then $p$ is [J] for $q$ and $p$ is [K] for $r$.
  3. if in condition $p$ we set $a=2$ we can transform the equation in $p$ into ${(x + \frac{b - [L]}{[M]}y)}^2 + ([N] - \frac{{(b-[O])}^2}{[P]})y^2 = 0$

hence $p$ is a necessary and sufficient for$q$ if and only if $b$ satisfies $[Q] < b < [R]$

we are to find $[H]$ until $[R]$

for $[H], [I], [J],[K]$ the are options to choose

$0$ necessary and sufficient condition

$1$ a necessary condition but not sufficient condition

$2$ a sufficient condition but not necessary condition

$3$ neither a necessary condition nor sufficient condition

i know some examples, for sufficient condition the example = boiling potato is a sufficient condition to cook it, but not necessary since there are many ways to cook potato. an example for necessary condition = if want one to go to college one must be a human, being a human is necessary condition to go to college but not sufficient. example for necessary and sufficient condition = for a building to be called a house is to have alive being to live it.

but how to apply it in these math equations.. any people who is good with this subjects? i am not good with this subject. or anyone can give me some hints where should i start?

  • $\begingroup$ What does "p is [H] for q" mean? (By the way, you should read the description of a tag before applying it - this has nothing whatever to do with functional analysis...) $\endgroup$ – David C. Ullrich Feb 14 at 15:07

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