Discrete Structures - Fall 1994 - Examination 1
1. (20 pts.) Circle T if the statement is true or F if the statement is false.
TF(R ∩ Q) ∪ Z = Z.
TFIf P(A) is the Power Set of A and ∅ is the Empty Set, then ∅ ⊂ P(A) and ∅∈ P(A).
T F The negation of the statement All students who take CMSC 203 and pass will get a
Computer Science degree is the statement All students who take CMSC 203 and
pass will not get a Computer Science Degree.
T F The statements If I pass CMSC 203, then I will get a good job and If I do not get a good
job, then I did not pass CMSC 203 are logically equivalent.
T F If A = {x,y,z}, then (z,z,y) ∈ A × A.
TFIf Σ = {x,y,z}, then (zz,y) ∈ Σ2 × Σ.
T F The set of the integers is partitioned by the set of positive integers and the set of negative
integers .
TF(Show work!) [[(33 mod 6) - ] ] mod 5 = 2 .
T F If n and d are positive integers, then n = d(n div d) + (n mod d).
T F If A is a set with 8 elements, and a ∈ A, then A has 128 subsets containing a.
2. (6 pts.) Given the statement All kids who run fast win races,
its converse is ____________; its inverse is___________; its contrapositive is_____________.
3. (10 pts.) Show, without using truth tables, that p → ( q → r ) ≡ ( p ∧ q ) → r.
4. (4 pts.) Write the negation of the universal statement: For all x ∈ Z, if x2 = 9, then x = 3 or x = −3.
5. (10 pts.) Find the Truth Table and Boolean Polynomial representing a circuit of three switches con-
trolling a light bulb in such a way that if the first and third switches are both ON or both OFF, the bulb is
ON.
6. (10 pts.) Using the Properties of Sets, show that: (A − B) − C = (A − C) − B.
7. (40 pts.) Prove 2 of the 4 theorems:
Theorem 1: If every integer has a unique representation as the product of primes, then there are an
infinite number of primes.
Theorem 2: If any integer n can be expressed as 3k, 3k + 1, or 3k + 2, then the product of three con-
secutive integers is divisible by 3..
Theorem 3: If q ∈ Q and r ∈ Q, then qr ∈ Q.
Theorem 4: For all integers a,b,c, and d, if a | (b + c) and a | d, then a | (b + c + d).
126
102
---------13.2–()6+
Docsity.com