Discrete Structures - Fall 1996 - Examination 1
1. (20 pts.) Circle T if the statement is true or F if the statement is false.
TFN ∪ Z = Q.
TF{1,2,3} ∈ P ({1,2,3,4}). (Note: P(A) denotes the Power Set of A.)
TFThe negation of the statement All students are motivated and not lazy is the statement
Some students are either not motivated, or are lazy, or both.
TFThe inverse of the statement If I pass CMSC 203, then I get a good job is:
I pass CMSC 203, but I do not get a good job.
TFIf A = {x,y,z}, then A × A = {(x,x),(x,y),(x,z),(y,x),(y,y),(y,z),(z,x),(z,y),(z,z)}.
TFIf Σ = {x,y,z}, then xyz ∈ Σ2.
TFThe sets {1,3,5}, {1,2,3}, and {3,4,5} partition the set {1,2,3,4,5}.
TFIf d divides n, then n/d = n mod d.
TFFor all integers x > 1, if x is prime, then x is odd.
TFA set with 8 elements has 255 non-empty subsets.
2. (6 pts.) Use the Euclidean Algorithm to find gcd(744,122).
3. (10 pts.) Find the truth table of the Boolean Polynomial F(x,y,z) = xy’ + (x’ + z)’.
4. (7 pts.) Show the following is a valid argument:
If x is not rational, then xy is real
z is an even integer and xy is real
x is rational or z is a prime integer
∴ z is a prime integer.
5. (10 pts.) Given the quantified statement: For all x ∈ Z, if x is odd, then (x + 1) is even ,
find its:
Converse:____________________________________________________________
Inverse:______________________________________________________________
Contrapositive:_________________________________________________________
Negation:_____________________________________________________________
6. (7 pts.) For the sets A = {1,2,3,4,5}, B = {2,4,6}, and C = {1,2,4,8} from the Universal Set
U = {0,1,2,3,4,5,6,7,8,9}, show that: (A ∩ Bc)c − (A ∩ B) = Ac.
7. (40 pts.) Prove 2 of the 4 theorems:
Theorem 1: For all integers n, if n3 is even, then n is even.
Theorem 2: is irrational. (Hint: Assume Theorem 1 is true.)
Theorem 3: If n and d are positive integers and (n div d) = (n mod d), then (d + 1) | n.
Theorem 4: For all non-zero integers a and b, if a | b and b | a, then a = b.
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