MAD 2104 Midterm: MAD 2104 FIU Exam 211k
MAD 2104 Sept 15, 2011
Quiz 2 and Key Prof. S. Hudson
1) [20pts] List all the elements of the power set P({1,2,3}):
2) [20pts] Find a compound proposition in disjunctive normal form equivalent to (p→q)∧r.
3) [30pt] Express the negation of each of these, so that all negation symbols appear just
before predicates (eg simplify these):
a) ∀x∃y(P(x, y)→Q(x, y))
b) ∀x∃y∀zT (x, y, z)
4) [30pt] Choose ONE proof. Use sentences (rather than Venn diagrams, etc).
a) Prove that √3 is irrational, using a proof by contradiction (very similar to the one
done in class for √2).
b) Prove for every integer n, that n2≥n. You may want to handle the cases of n > 0
and n < 0 separately, and may even need to include other case(s), such as n= 0 and n= 1,
etc. You can use any basic algebra to do this; I am mainly interested in your organization
of the proof.
c) [This is related to Ch 2.2, but you don’t have to choose it!] Prove that A∪(B∩C)⊆
(A∪B)∩(A∪C).
Tiny Bonus [about 2 pts]: Who said ‘I am not a robot. I am a unicorn.’ ?
Remarks and Answers: The average was about 68 / 100. The unofficial scale is
A’s 80-100
B’s 70-79
C’s 60-69
D’s 50-59
1) List the 23= 8 subsets, using correct notation for sets;
∅,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}
Note that 1 6={1}and (1,2) 6={1,2}, though I usually gave partial credit for such answers
if the intention seemed clear.
2) Use a truth table as described in exercise 1.2.42 [and in class] to find the 3 conjunctions
needed;
(p∧q∧r)∨(¬p∧q∧r)∨(¬p∧ ¬q∧r)
1
Document Summary
You can use any basic algebra to do this; i am mainly interested in your organization of the proof: [this is related to ch 2. 2, but you don"t have to choose it!] Prove that a (b c) (a b) (a c). Tiny bonus [about 2 pts]: who said i am not a robot. Remarks and answers: the average was about 68 / 100. D"s 50-59: list the 23 = 8 subsets, using correct notation for sets; Most people reasoned that out, and some others used a truth table. I did not insist on seeing much work on this problem. 4a) probably this one is harder than the others, even though we did a similar proof in class, and a hint was given on the board. The rst line should be something very very close to - assume 3 is rational, to get a contradiction. 4b) this is example 3 pg 88, not very hard, and the most popular choice.
