MATH1004_Discrete Math_2012 Semester 2_q_pract1_ans12
The University of Sydney
MATH1004
Practice questions for Quiz1
1.What is the number of balanced strings of5left and5right brackets which
end with RR?
Answer:28.Hint:28=42?14.
2.What is the number of planar diagrams with10dots without a“big smile”;
that is,without the arc connecting the?rst and the last dots?
Answer:28.Hint:28=42?14.
3.Find the symmetric di?erence P+Q of the sets P={1,{1,2},3,4}and Q=
{2,3,4}.
Answer:{1,{1,2},2}
4.Find the number of permutations of the set{1,2,3,4,5,6}such that the image
of the element1is odd.
Answer:360.
5.Let Z be the set of all integers and A={x∈Z|x4≤100}.Write down all
elements of A.
Answer:A={?3,?2,?1,0,1,2,3}.
6.Let A={a,b,c,d,1,2,3,4}and B={a,b,u,v,1,2,5,6,7}.Find the cardinal-
ities of the sets A∩B,A∪B,A\B,B\A,A+B.
Answer:4,13,4,5,9.
https://www.wendangku.net/doc/159704288.html,e the set theory notation to write down the set of all positive integers which
are divisible by7but not divisible by3.
Answer:{x|x=21k+7,k∈N}∪{x|x=21k+14,k∈N}.
8.Which of the following diagrams represent functions?
(a)(b)(c)(d)
Answer:(c).
9.Which of the functions in Question8are(i)injective?(ii)surjective?
Answer:The function represented by(c)is neither injective,nor surjective.
10.Let A={a,b,c,d}and B={1,2,3,4,5}.Which of the following sets of
ordered pairs represents a function from A to B?
(a){(a,1),(a,2),(a,3),(a,4)}(b){(a,2),(b,2),(c,3),(d,4)}
(c){(a,2),(b,2),(d,4)}(d){(a,1),(a,2),(b,3),(c,4),(d,5)}
Answer:(b).
11.We are told that f:A→B is an injective function,|A|=7and|B|≤9.
What are possible cardinalities of B?
Answer:7,8,9.
12.We are told that f:A→B is a surjective function,|A|=7and|B|≥6.What
are possible cardinalities of B?
Answer:6,7.
13.The function f:N→N is de?ned by f(n)=n+5.Is this function(i)injective?
(ii)surjective?(iii)bijective?
Answer:f is injective,but not surjective and not bijective.
14.The function f:Z→Z is de?ned by f(n)=n?5.Is this function(i)injective?
(ii)surjective?(iii)bijective?
Answer:f is injective,surjective and bijective.
15.The function f:N→Z is de?ned by f(n)=n?5.Is this function(i)injective?
(ii)surjective?(iii)bijective?
Answer:f is injective,but not surjective and not bijective.
16.The function f:Z→N is de?ned by f(n)=|2n|.Is this function(i)injective?
(ii)surjective?(iii)bijective?
Answer:f is not injective,not surjective and not bijective.
17.The function f:N→N is de?ned by f(n)=n+2;the function g:N→N is
de?ned by g(n)=2n.Write the formulas for h=g?f and k=f?g.
Answer:h(n)=2n+4,k(n)=2n+2.
18.The function f:Z→Z is de?ned by f(n)=|n?2|;the function g:Z→Z is
de?ned by g(n)=n+5.Write the formulas for h=g?f and k=f?g.
Answer:h(n)=|n?2|+5,k(n)=|n+3|.
19.How many strings of three digits are there if0is not used?
Answer:729.
20.Write down all elements of the set X×Y,where X={?,?,?}and
Y={?,?,?,?}.
Answer:
{(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?),(?,?)}.