Question 1

Contrapositive of P implies Q.

Accepted Answer

~Q implies ~P.

Question 2

Converse of P implies Q.

Accepted Answer

Q implies P.

Question 3

Modus Ponens.

Accepted Answer

If we know x and we know x implies y we can infer y.

Question 4

A is a subset of B iff

Accepted Answer

for all x: x in A implies x in B

Question 5

A is equal to B iff

Accepted Answer

for all x: x in A iff x in B

Question 6

union

Accepted Answer

x in A or x in B

Question 7

intersection

Accepted Answer

x in A and x in B

Question 8

tautology

Accepted Answer

a statement which is always true

Question 9

contradiction

Accepted Answer

a statement which is always false

Question 10

interval

Accepted Answer

for all x,y,z in R: x in I, z in I, x<y, y<z implies y in I

Question 11

open interval

Accepted Answer

for all y in I there exists x,z in I with y<z and x<y

Question 12

injective function

Accepted Answer

for all x,y in X: f(x)=f(y) implies x=y

Question 13

surjective function

Accepted Answer

for all y in Y there exists x in X such that f(x)=y

Question 14

bijective function

Accepted Answer

injective and surjective

Question 15

range of f(x)

Accepted Answer

{y in Y: there exists x in X s.t. f(x)=y}

Question 16

infinite set S is countable if

Accepted Answer

there exists a bijection f: N to S

Question 17

preimage of B

Accepted Answer

{x in X: f(x) in B}

Question 18

image of A

Accepted Answer

f(A)={y in Y: there exists x in A s.t. f(x)=y}

Question 19

composition g@f(x)

Accepted Answer

g(f(x))

Question 20

limit (x to infin)

Accepted Answer

f(x) to L as x to infin if
(for all E>0)(there exists k>0)(for all x in X) x>k implies |f(x)-L|<E

Question 21

cauchy criterion (x to infin)

Accepted Answer

f(x) to L as x to infin if
(for all E>0)(there exists k>0)(for all x1,x2 in X) x1,x2>k implies |f(x1)-f(x2)|<E

Question 22

limit (x to a)

Accepted Answer

f(x) to L as x to a if (for all E>0)(there exists d>0)(for all x in X) 0<|x-a|

Question 23

cauchy criterion (x to a)

Accepted Answer

f(x) to L as x to a if (for all E>0)(there exists d>0)(for all x1,x2 in X) 0<|x-a|

Question 24

continuous (simple)

Accepted Answer

f is continuous at a if f(x) to f(a) as x to a

Question 25

continuous (exact)

Accepted Answer

f is continuous at a if a in X and (for all E>0)(there exists d>0)(for all x in X) |x-a|

Question 26

Intermediate Value Theorem

Accepted Answer

if f is continuous on the closed interval [a,b] and f(a),f(b) have opposite signs then there exists c in (a,b) such that f(c)=0

Question 27

x in interior of S

Accepted Answer

if there exists an open interval (a,b) in S with x in (a,b)

Question 28

min and max

Accepted Answer

if f is a continuous function on a closed interval [a,b] then f achieves its min and max (Cmin, Cmax in [a,b]) such that f(Cmin)<eq f(x)<eq f(Cmax) for all x in [a,b]

Question 29

differentiable

Accepted Answer

at a if a in X and [f(a+h)-f(a)]/h to a limit as h to 0
continuous

Question 30

a in domain of f then f is diff. at a with derivative m iff

Accepted Answer

there exists e(x) continuous at a with e(a)=0 such that f(x)=f(a)+(m+e(x))(x-a)

Question 31

f'(c)=0 if

Accepted Answer

f has local min/max at c

Question 32

Mean Value Theorem

Accepted Answer

if f is continuous on [a,b] and diff on (a,b) then there exists c in (a,b) s.t. f'(c)=(f(b)-f(a))/(b-a)

Question 33

increasing

Accepted Answer

for all a,b if a<b then f(a)<eq f(b)

Question 34

strictly increasing

Accepted Answer

for all a,b if a<b then f(a)<f(b)

Question 35

e as x to infin

Accepted Answer

lim (1+(1/x))^x

Question 36

f is twice diff on (a,b) and f(x),f'(x) continuous on [a,b]

Accepted Answer

there exists c in (a,b) s.t. f(b)=f(c)+f'(c)(b-a)+f''(c)(.5)(b-a)^2

Question 37

nth Taylor polynomial

Accepted Answer

is f is n times diff at a then Pn(x)= sum from m=o to n
{[f^m(a)]/m!}(x-a)^m

Question 38

partition of [a,b]

Accepted Answer

a list a0,a1,..,an where a0=a,
a0<a1<..<an and an=b

Question 39

area under f

Accepted Answer

f(x1)(a1-a0)+f(x2)(a2-a1)+..+f(xn)(an-a{n-1}) for some points x1,x2,..,xn with x1 in (a0,a1), x2 in (a1,a2),..,xn in (a{n-1},an)

Question 40

f has integral A on [a,b] if

Accepted Answer

(for all E>0)(there exists partition a0,a1,..,an of [a,b])(for all x1,x2,..,xn) if x1 in (a0,a1), x2 in (a1,a2),.., xn in (a{n-1},an) then mod
sum from m=1 to m=n of
f(xm)(am-a{m-1}) - A
is less than E

Question 41

f integrable on [a,b]

Accepted Answer

integrated from b to a: f(x)dx = A
continuous

Question 42

fundamental theorem of calculus (i)

Accepted Answer

if f continuous on [a,b] then F(x)=
integrated from x to a: f(t)dt is diff on [a.b] and F'(x)=f(x)

Question 43

fundamental theorem of caluculus (ii)

Accepted Answer

if F diff on [a,b] and inverseF(x) continuous on [a,b] then
integrated from b to a: inverseF(X)dx = F(b)-F(a)

MATH1050

University of Southampton MATH1050

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