Core 1 Proof

Core 1 Proof - Transcript

CORE 1
proof
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What is Mathematical Proof?
The  p ro c e s s  o f s ta rting  with  a n 
a s s um ptio n, o r a  s ta te m e nt whic h  is  
g ive n, a nd , b y us ing  lo g ic a l a rg um e nt, 
a rriving  a t a  c o nc lus io n

Mathematical Proof
‘P ro ve  th a t …’ o r ‘G ive n  …, p ro ve  …’ o r ‘ P ro ve  …, g ive n …’
Fo rm  a  lo g ic a l a rg um e nt
S ta rt with  wh a t is  g ive n  o r s ta nd a rd  re s ults
De d uc e  e a c h  s te p  fro m  p re vio us
S ta nd a rd  re s u lts  c a n b e  us e d  a t a ny  s ta g e

MATHEMATICAL STATEMENTS
ΔABC is isosceles
sinθ = ¾
The gradient of y=mx+c is m

Us e  to  e xp re s s  th e  re la tio ns h ip  b e twe e n  s ta te m e nts
im p lie s
d o e s  no t im p ly
is  im p lie d  b y
im p lie s  a nd  is  im p lie d  b y
Implication Signs

Example:
Prove that ΔABC is isosceles
AB = AC
‗B = _C
AB = AC ΔABC is isosceles
A
B C

Example:
Link the statements a = 0 and ab = 0 using implication signs.
a = 0 ab = 0
ab = 0 a = 0 (b could be 0)
ab = 0 Either a = 0 or b = 0

Example:


Example:
Pro ve  tha t s um  o f a n e ve n num b e r a nd  
a n o dd  num b e r is  a lwa ys  o dd .

Let 2n be any even number, where n is an integer.
2m + 1 be any odd number, where m is an integer.
2n + 2m + 1 = 2(n+m) + 1
n+m is an integer
n+m is an integer
2(n+m) is even
2(n+m) + 1 is odd
2(n+m) + 1 = 2n + 2m + 1
the sum of an even number and an odd number is
always odd.