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V 76%! Wiles MATH 135, Fall 2010 Solution of Assignment #1 Problem 1. Find all ordered pairs of integers (ac, y) such that x2 + 23: + 18 = 3/2. SOLUTION. Note that 17=(y+x+1)(y*9341) Since y + ac + 1 and y — a; — 1 are integers, there are only the following possibilities
y+m+1:i17,ymm—1:i1,ory+m+1=i1,y—m—1=i17.®
The solutions are
y=9,x=7, ory=—9,a:=—9, ory=9,ac:—9, ory=—9,m=7. I @
Problem 2. Find all real numbers a: such that 113—13 “2\/E«1
ﬁ+3_ 3 ' SOLUTION.
a: — 13 _ 2\/E — 1 ﬂ + 3 " 3
Let ﬂ = y. The equation becomes y2—5y—~36=O => (y—9)(y+4):0 => 21:91—4 =>3$—39=2$+5\/a?—3=>$—5\/_—36=0. M > Since x is a real number, y : ﬂ must be positive. Thus, y : 9 and a: = 81. I (D
M Problem 3. Solve y : a: —— % with 0 < :2: S 1 for a: in terms of y. SOLUTION. Multiply both sides of the equation by a: then use the quadratic formula to get 1 :i:\/2—4
y=m+—<=>$y=m2+1<=>x2—Ty+1—0<—>xy 2y . a:
Notethat0<$31 $21220 y=x+%22x => $S%,sowemustusethe _ T (Q
$:y_+_4. Problem 4. Let P and Q be statements. Show that the statements NOT (P QR Q) and
(NOT P) AND (NOT Q) have the same truth tables and give an example of the equivalence of
these statements in everyday language. SOLUTION. NOT (P OR Q) P Q NOTP NOTQ (NOT P) AND (NOT Q) T T F F F g)
T F F T F F T T F F F F T T T cabbage or broccoli”. This means that “I do not want cabbage” and “I do not want broccoli”. I The ﬁnal colu s of each table are the same so tatements have the same truth tables.
This equivalence can e illustrated in everyday language. Consider t e s a emenWITdBAniot‘W'ant] Problem 5. Let P, Q, and R be statements. Show that the statements P AND (Q OR R) and
(P AND Q) OR (P AND R) have the same truth tables; This is a distributive law. SOLUTION. H T
T
F
T
T
T
F AND—21 P AND R (P AND Q) OR (P AND R) @ hidianHTar—BHD
*Tl’ijlEH’UWHHQ
mewawamam
ww'ﬁ'ﬁhjhjﬂﬂ
mwwmeHw—a
ninjqu’ijﬁrBH The ﬁnal columns of each table are the same, so the two statements have the same truth tables.  ...
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 2$, $31, emenWITdBAniot‘W'ant, $21220

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