University of Southampton MATH1050
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| Contrapositive of P implies Q. | ~Q implies ~P.
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| Converse of P implies Q. | Q implies P.
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| Modus Ponens. | If we know x and we know x implies y we can infer y.
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| A is a subset of B iff | for all x: x in A implies x in B
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| A is equal to B iff | for all x: x in A iff x in B
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| union | x in A or x in B
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| intersection | x in A and x in B
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| tautology | a statement which is always true
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| contradiction | a statement which is always false
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| interval | for all x,y,z in R: x in I, z in I, x<y, y<z implies y in I
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| open interval | for all y in I there exists x,z in I with y<z and x<y
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| injective function | for all x,y in X: f(x)=f(y) implies x=y
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| surjective function | for all y in Y there exists x in X such that f(x)=y
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| bijective function | injective and surjective
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| range of f(x) | {y in Y: there exists x in X s.t. f(x)=y}
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| infinite set S is countable if | there exists a bijection f: N to S
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| preimage of B | {x in X: f(x) in B}
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| image of A | f(A)={y in Y: there exists x in A s.t. f(x)=y}
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| composition g@f(x) | g(f(x))
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| limit (x to infin) | 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
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| cauchy criterion (x to infin) | 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
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| limit (x to a) | 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|<d implies |f(x)-L|<E
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| cauchy criterion (x to a) | 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|<d and 0<|x-a|<d implies |f(x1)-f(x2)|<E
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| continuous (simple) | f is continuous at a if f(x) to f(a) as x to a
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| continuous (exact) | 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|<d implies |f(x)-f(a)|<E
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| Intermediate Value Theorem | 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
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| x in interior of S | if there exists an open interval (a,b) in S with x in (a,b)
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| min and max | 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]
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| differentiable | at a if a in X and [f(a+h)-f(a)]/h to a limit as h to 0
continuous
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| a in domain of f then f is diff. at a with derivative m iff | there exists e(x) continuous at a with e(a)=0 such that f(x)=f(a)+(m+e(x))(x-a)
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| f'(c)=0 if | f has local min/max at c
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| Mean Value Theorem | 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)
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| increasing | for all a,b if a<b then f(a)<eq f(b)
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| strictly increasing | for all a,b if a<b then f(a)<f(b)
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| e as x to infin | lim (1+(1/x))^x
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| f is twice diff on (a,b) and f(x),f'(x) continuous on [a,b] | there exists c in (a,b) s.t. f(b)=f(c)+f'(c)(b-a)+f''(c)(.5)(b-a)^2
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| nth Taylor polynomial | is f is n times diff at a then Pn(x)= sum from m=o to n
{[f^m(a)]/m!}(x-a)^m
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| partition of [a,b] | a list a0,a1,..,an where a0=a,
a0<a1<..<an and an=b
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| area under f | 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)
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| f has integral A on [a,b] if | (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
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| f integrable on [a,b] | integrated from b to a: f(x)dx = A
continuous
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| fundamental theorem of calculus (i) | 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)
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| fundamental theorem of caluculus (ii) | 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)
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