COMP-047 Discrete mathematics Exam

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COMP-047 Exam #1

Name:
______________________________________________________________________________

1) (6 pts) Show that (p ?q) ? ( ~q ?  ~p) a tautology.

2) (4 pts) If A = {1,2 3,4} and B = {a,b} what is the cardinality of the powerset of AXB?

3) (7 pts) Give a direct proof of the following: If x,y  are integers and y is odd, then 2x + y  + 1 is  even

4) (5 pts) Consider the set A = {1,2 3}  and B = { a,b,c,d} and the function f: A ? B. What is the cardinality of f?

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COMP-047 Exam #1 Assigned Tuesday Feb 26 Due 8 March Name: ______________________________________________________________________________ 1) (6 pts) Show that (p →q) ↔ ( ~q → ~p) a tautology. 2) (4 pts) If A = {1,2 3,4} and B = {a,b} what is the cardinality of the powerset of AXB? 3) (7 pts) Give a direct proof of the following: If x,y are integers and y is odd, then 2x + y + 1 is even 4) (5 pts) Consider the set A = {1,2 3} and B = { a,b,c,d} and the function f: A → B. What is the cardinality of f? 5) (4 pts) Rewrite the following using universal quantification. Universally Qualified Statement ¬∃y G(Bill, y) . ¬∃x (Q(x) ⋁ F (x)). 6) (7 pts) Without using a truth table explain whether the following statement is a tautology, a contradiction or a continency: ( a ⋀ ¬b) ⋀ (¬a ⋁ c) ⋀ (¬c ⋁ b ) 7) ( 7 pt s ) Is ( ( p→r) ⋁ ( q →r) ) ↔ ( ( p ⋀ r ) →r ) a tautology, a contradiction or a continency. 8) (7 pts) Give a proof by contradiction of the following: : If x,y are integers and y is odd, then 2x + y + 1 is even. 9) (6 pts) Given a three element set A: {a1, a2, a3} and a two element set set B: {b1, b2}. Enumerate all the mappings f: A→B. 10) (7 pts) Use rules of inference to show If the following premises are true 1. 2. 3. 4. J i l l is a Math Major or a Physics Major. If one does not know math, one is not a Math Major. If one knows math, one is employable. J i l l is not a Physics Major. Let M(X) be X is a Math Major Let P(X) be X is a Physics Major Let K(X) be X knows Math Let E(X) be X is Employable Then show that the conclusion ”Jill is employable” is true (i.e. E(Jill) is true). Step Reason 11) (5 pts) If h(x) is x2+1 and g(x) = (x-1).5 then what is (g◦h)(x) 12) (8 pts) Prove that if n is an integer, then ⌊(𝑛/2)⌋ is n/2 if n is even and (n-1)/2 if n is odd. 13) (8 pts) Prove the second of De Morgan’s Laws holds by showing that if A and B are sets, then ̅̅̅̅̅̅̅ 𝐴 ∪ 𝐵 = 𝐴̅ ∩ 𝐵̅ by showing that each side is a subset of the other. 14) (6 pts) Without using a truth table, show that ~ ( p ⋁ (( ~p ) ⋀ q) ) ↔ (~p) ⋀ (~q) 15) (6 pts) Suppose that P(m, n) means ”m ≤ n”, where m and n is the set of positive integers. What is the truth value of the statement? • ∀nP(0, n). • ∃m∀nP(m, n). • ∀m∃nP(m, n). 16) (7 pts) Use rules of inference to show that if the premises: p∨q→r s ∨ ¬q t ¬p → ¬t p ∧ r → ¬s are true, then the conclusion ¬q is true. Extra Credit: (5 pts) Prove or Disprove ⌊√ ⌊𝑥⌋ ⌋ = ⌊√ 𝑥 ⌋ ...
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Running head: DISCRETE MATHEMATICS EXAM
1

Discrete mathematics
Name of Student
Name of Institution

DISCRETE MATHEMATICS EXAM
1. Show that (p →q) ↔ (~q → ~p) a tautology. (6 points)
De Morgan’s laws states that intersection of two complement sets is similar union of two sets
complement and vice versa.
Therefore,
Not A or nor B is the same to not (A and B)
Not A and Not B is the same as not (A or B)
In V=Boolean algebra and set theory it is written as

Where
A and B are sets
A compliment A
∪ is the union.
∩ is the intersection
Rules written in formal language

And

Where
P and Q are propositions,
(ᴧ) (AND) is the conjunction logic operator,
(¬) (NOT) is the negation logic operator,
(˅) (OR) is the disjunction logic operator,

2

DISCRETE MATHEMATICS EXAM

3

2. If A = {1,2 3,4} and B = {a,b} what is the cardinality of the powerset of AXB?(4 pts)
Cardinally refers to the number of elements in a finite set and power set of A or P (A). Hence,
cardinality of P (A) refers to the number of subsets of A.

Here, A=0,1,2,3,4,5,6,7,8,9,10

|A|
I.e. cardinality of A is 11.

If A
Is a finite set with |A| = n elements, then the number of subsets of A is=2n.

Again, |P (A)|
= number of subsets of A=2n

Putting n=11
, we get that |P (A)| is 211
3. Give a direct proof of the following: If x,y are integers and y is odd, then 2x + y + 1 is
even (7 pts)
x,y are integers, and y is odd
2x + y + 1 is even
x and y are integers
and y is odd, set y be 2n+ 1
x can either be odd or even
But
2x+y+1= ˃ here 2x is even since x is multiplied by 2

DISCRETE MATHEMATICS EXAM

4

y is odd, then y+1 is even
2𝑋
𝑒𝑣𝑒𝑛

+

𝑦+1
𝑒𝑣𝑒𝑛

=˃Adding two even numbers is again an even

Hence proved
4. Consider the set A = {1,2 3} and B = { a,b,c,d} and the function f: A → B. What is the
cardinality of f?(5 pts.)
A = {1, 2, 3}
B = {a, b, c, d}
f: A -˃ B
Cardinality of f = cardinality of A
= A (number of elements A)
=3
5. Rewrite the following using universal quantification. (4 pts)
Universally Qualified Statement
We assume the exp...

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