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[answered] MATH 109 WINTER 2015 FINAL EXAM Instructions: There are 85


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1 (15 pts). (short answers)(a) (4 pts). Are the following two propositional statements logically equivalent? Justifyyour answer with a truth table.(i) if (not P), then Q(ii) P or Q(b) (5 pts). What is the remainder when 31000 is divided by 10? Briefly ustify your answerusing congruence modulo 10.(c) (3 pts). Let X, Y, Z be subsets of a set U. Write down the inclusion/exclusion formulawhich calculates the size of |X ? Y ? Z|.(d) (3 pts). State the pigeonhole principle.2 (10 pts).(a) (3 pts). State the division theorem.(b) (7 pts). Prove that for all integers n ? Z, 3|n if and only if 3|n2.3 (10 pts). Prove that Xni=11i(i + 1) =nn + 1for all integers n ? 1.4 (10 pts).(a) (5 pts). Find all solutions to the equation 15x + 21y = 12 with x, y ? Z. Justify youranswer in a few sentences by quoting the main theorems on the solution to linear diophantineequations.(b) (5 pts). Find all solutions to the conguence 21y ? 12 (mod 15) with y ? Z, by usingyour answer to part (a). Give a few sentences justification explaining why your answerfollows from part (a).Date: March 16, 2015.15 (10 pts). Let f : X ? Y and g : Y ? Z be functions. Suppose that the compositiong ? f : X ? Z is a surjective function.(a) (5 pts). Must f be surjective? Either prove it must be or give a counterexample.(b) (5 pts). Must g be surjective? Either prove it must be or give a counterexample.6 (10 pts).(a) (5 pts). Let X = R2 = {(a, b)|a ? R, b ? R} be the set of all ordered pairs of realnumbers, in other words the Cartesian plane. Define a relation on X by (a, b) ? (c, d) if andonly if b ? d = a2 ? c2. Prove that ? is an equivalence relation on A.(b) (5 pts). Describe geometrically what the equivalence classes of ? and the correspondingpartition of R2look like. Sketch some graphs as part of your answer.7 (10 pts). Let a and b be positive integers. Recall that the least common multiple of aand b is the unique positive integer m such that (i) a|m and b|m, and (ii) given any integern with a|n and b|n, then m ? n.Let m be the least common multiple of a and b. Prove that if n is any positive integersuch that a|n and b|n, then m|n.8 (10 pts). Let X and Y be sets and let f : X ? Y be a function. Recall that for any subsetA ? X, the image of A under f is defined to be f(A) = {y ? Y |y = f(a) for some a ? A}.(Your textbook uses the notation?f (A) for f(A).)(a) (5 pts). Prove that for any subsets A1 ? X, A2 ? X,f(A1 ? A2) ? f(A1) ? f(A2).(b) (5 pts). With the same setup as in part (a), prove that if f is an injective function,then f(A1 ? A2) = f(A1) ? f(A2).


MATH 109 WINTER 2015 FINAL EXAM

 

Instructions: There are 85 points total. Justify all of your answers, and show all of your

 

work in your blue book. You may use the result of one part of a problem in the proof of a

 

later part, even if you do not complete the proof of the earlier part. You may quote basic

 

theorems proved in the textbook or in class, unless the problem says otherwise, but do not

 

quote the results of homework exercises. 1 (15 pts). (short answers)

 

(a) (4 pts). Are the following two propositional statements logically equivalent? Justify

 

your answer with a truth table.

 

(i) if (not P), then Q

 

(ii) P or Q

 

(b) (5 pts). What is the remainder when 31000 is divided by 10? Briefly ustify your answer

 

using congruence modulo 10.

 

(c) (3 pts). Let X, Y, Z be subsets of a set U . Write down the inclusion/exclusion formula

 

which calculates the size of |X ? Y ? Z|.

 

(d) (3 pts). State the pigeonhole principle.

 

2 (10 pts).

 

(a) (3 pts). State the division theorem.

 

(b) (7 pts). Prove that for all integers n ? Z, 3|n if and only if 3|n2 .

 

3 (10 pts). Prove that n

 

X

 

i=1 n

 

1

 

=

 

for all integers n ? 1.

 

i(i + 1)

 

n+1 4 (10 pts).

 

(a) (5 pts). Find all solutions to the equation 15x + 21y = 12 with x, y ? Z. Justify your

 

answer in a few sentences by quoting the main theorems on the solution to linear diophantine

 

equations.

 

(b) (5 pts). Find all solutions to the conguence 21y ? 12 (mod 15) with y ? Z, by using

 

your answer to part (a). Give a few sentences justification explaining why your answer

 

follows from part (a).

 

Date: March 16, 2015.

 

1 5 (10 pts). Let f : X ? Y and g : Y ? Z be functions. Suppose that the composition

 

g ? f : X ? Z is a surjective function.

 

(a) (5 pts). Must f be surjective? Either prove it must be or give a counterexample.

 

(b) (5 pts). Must g be surjective? Either prove it must be or give a counterexample.

 

6 (10 pts).

 

(a) (5 pts). Let X = R2 = {(a, b)|a ? R, b ? R} be the set of all ordered pairs of real

 

numbers, in other words the Cartesian plane. Define a relation on X by (a, b) ? (c, d) if and

 

only if b ? d = a2 ? c2 . Prove that ? is an equivalence relation on A.

 

(b) (5 pts). Describe geometrically what the equivalence classes of ? and the corresponding

 

partition of R2 look like. Sketch some graphs as part of your answer.

 

7 (10 pts). Let a and b be positive integers. Recall that the least common multiple of a

 

and b is the unique positive integer m such that (i) a|m and b|m, and (ii) given any integer

 

n with a|n and b|n, then m ? n.

 

Let m be the least common multiple of a and b. Prove that if n is any positive integer

 

such that a|n and b|n, then m|n.

 

8 (10 pts). Let X and Y be sets and let f : X ? Y be a function. Recall that for any subset

 

A ? X, the image of A under f is defined to be f (A) = {y ? Y |y = f (a) for some a ? A}.

 

? (Your textbook uses the notation f (A) for f (A).)

 

(a) (5 pts). Prove that for any subsets A1 ? X, A2 ? X,

 

f (A1 ? A2 ) ? f (A1 ) ? f (A2 ).

 

(b) (5 pts). With the same setup as in part (a), prove that if f is an injective function,

 

then f (A1 ? A2 ) = f (A1 ) ? f (A2 ). 2

 


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