Answers & Hints for Selected Exercises 703
- :Jx, 3y 3 [S(x,y) /\"' C(x,y)]; domain of x and y ={triangles}
- :Jx 3 Vy, [(y "I-x) =? x < y]; domain of x and y ={numbers in this set}
- "'3x 3 Vy, x ?". y or "ix, 3y 3 y > x; domain of x and y ={real numbers}
Part B :
- :Jx 3 "' [L(x) =? W(x)] or :Jx 3 [L(x) /\ "' W(x)]. Not all lawyers are
wealthy. Or, some lawyers are not wealthy.
- 3x 3 [A(x) /\ M(x)]. There is someone who wants an A in this course and
can afford to miss an assignment.
- "' [:Jx 3 G(x)] V "'"' [:Jx 3 C(x)] or [Vx, "'G(x)] V [:Jx 3 C(x)]. Either no
one in this room is guilty, or someone in this room will be charged.
- "ix, "' [L(x) /\"' G(x)] or "ix, ["' L(x) V "'"' G(x)] or "ix, [L(x) =? G(x)].
Everyone waiting in line for the show will get in.
- 3x 3"-' [G(x) =? C(x)] or :Jx 3 [G(x) /\ "'C(x)]. There is someone in the
room who is guilty but need not confess now.
- :Jx 3 x^2 + 3x - 1 = 0. The equation x^2 + 3x - 1 = 0 has a real number
solution.
- "ix, A(x). I can agree with all of your ideas.
15."' [Vx, E(x)] V"' ["-'"ix, S(x)] or [:Jx 3 "'E(x)]v[Vx, S(x)J. Either someone
is not eligible to try, or everyone will succeed.
- :Jx 3 "' [E(x) =? 3y 3 "' H(y)] or :Jx 3 [E(x) /\ "' 3y 3 "' H(y)] or
3x 3 [E(x) /\Vy, H(y)]. There is someone who could get elected and with
whom everyone would be happy.
- :Jx, 3y 3 "'B(x, y). There exists a pair of men who are not brothers.
- :Jx, 3y 3 [x "I-y /\ L(x, y)]. There are two people who look exactly alike.
- "ix, Vy, "' [S(x, y) /\ "' C(x, y)] or "ix, Vy, ["-' S(x, y) V C(x, y)] or Vx,Vy,
[S(x, y) =? C(x, y)J. If triangles are similar, they are congruent.
- "ix, 3y 3 "' [(y "I-x) =? x < y] or "ix, 3y 3 [(y "I-x) /\ x ?". y]. For every
member of this set there is a number in the set smaller than it. (No number in
the set is smaller than all the rest.)
27. :Jx 3 Vy, x ?". y. There is a largest real number.
Part C:
- (a) For every integer there is an odd integer t hat, when added to it, yields
an even sum. False: t ake n = 0.
