7 5 points Let U be the set of all circles in R2 having cent

7. (5 points) Let U be the set of all circles in R2 having center the origin. Interpret the origin being in this set; a circle centered at the origin with radius zero. Let Ci be elements of U. Let Ch +ca be the circle centered at the origin, whose radius is the sum of the radii of C, and ca. Let kci centered at the origin, whose radius times that of C1. The set U be the circle Determine which you is space axioms (listed below) hold and which do not. For each axiom that does not hold, provide justification. (Hint: more than one does not hold.) claim 1. Closed under addition; that is, the sum u v exists and is an element of U. 2. Closed under scalar multiplication; that is, cu s an element of U. 3. u +v v u. 4. u (v w) (u v) w 5. There exists an element of U, called a zero vector, denoted 0, such that u 0 u. 6. For every element u of U, there exists an element called a negative of u, denoted u, such that u (-u) 0. 7. c (u v) cu cv 8.. (c d)u cu du 10, lu u

Solution

The axioms at Sr. No. 6 does not hold as there is no concept of a circle with a negative radius. It does not exist.