B gandalf fools everyone c the only person frodo can

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(b) Gandalf fools everyone. (c) The only person Frodo can fool is himself. _________________________________________________________________ 5. (5 pts.) Determine the truth value of each of the following statements if the universe of discourse of each variable is the set of natural numbers, = {0,1,2,...}. (a) ( x)( y)(x + y = y) (b) ( x)( y)(x + y = y) _________________________________________________________________ 6. (5 pts.) If A is a countable set and B is an uncountable set, must A - B be countable? Briefly explain.
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TEST1/MAD2104 Page 3 of 4 _________________________________________________________________ 7. (15 pts.) Suppose A = { ,{ },2} and B = {2,3}. Then A B = A × B = |P(A)| = _________________________________________________________________ 8. (5 pts.) Suppose A and B are subsets of a universal set U. Show that A - B = A B. [Hints: (1) x ε A x ∉ ∅ → ... . (2) A - B = ∅ → ∀ x( x ε A - B x ε ∅ ). The contrapositive of the implication within the parentheses here is useful in dealing with the ellipsis in hint #1.] _________________________________________________________________ 9. (5 pts.) If f:X Y is a function, f -1 may be used to denote two quite different things. What are they? [Use complete sentences.]
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TEST1/MAD2104 Page 4 of 4 _________________________________________________________________ 10. (15 pts.) Suppose that f: → Ζ is the function defined by the formula f(x)= x , and suppose that A = {x ε | -3 x 3} and B = {x ε | -1 < x π }. Using appropriate notation, give each of the following. A - B = f(B) = f -1 ({1,3}) = _________________________________________________________________ 11. (5 pts.) What is the value of the following sum of terms of a geometric progression? [Hint: You may wish to re-index the varmint.] 8 2 j = j=1 _________________________________________________________________ 12. (5 pts.) Suppose g:A B and f:B C are functions. Prove exactly one of the following propositions. Indicate clearly which you are demonstrating. (a) If f g:A C is injective, then g is injective. (b) If f g:A C is surjective, then f is surjective.
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