Let k, be a theory whose language has only = as a predicate letter and no function letters or individual constants. let its proper axioms be (vx1)x1 = x1, (vxl)(vx2)(x2 = x2 * x2 = xı), and (vx1)(vx2)(vx3)(x1 = x2 = (x2 = xz x1 = x; show that k is a theory with equality. (hint: it suffices to prove that + xy = x3 = (x2 = x2 = x3 = x,) and x2 = x3 = (x1 = x2 = x2 = x3).] k, is called the pure first-order theory of equality.

Triangle xyz, with vertices x(-2, 0), y(-2, -1), and z(-5, -2), undergoes a transformation to form triangle x? y? z? , with vertices x? (4, -2), y? (4, -3), and z? (1, -4). the type of transformation that triangle xyz undergoes is a . triangle x? y? z? then undergoes a transformation to form triangle x? y? z? , with vertices x? (4, 2), y? (4, 3), and z? (1, 4). the type of transformation that triangle x? y? z? undergoes is a .