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# Proofs Post 6

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Proofa bya contradictiona anda proofa bya contrapositivea area similara ina aa
waya thata theya botha createa aa statementa toa eithera provea aa hypothesisa toa
bea truea ora false,a buta theya differa ina thea waya thata thea statementsa area
created.a Proofa bya contrapositivea beginsa witha thea negationa ofa thea
conclusiona anda provinga thata thea negationa ofa thea hypothesisa isa true.a
Proofa bya contradiction,a ona thea othera hand,a takesa thea negationa ofa thea
theorema toa showa thata inconsistenciesa willa arise,a provinga thea originala
theorema toa bea true.a Below,a Ia willa providea proofa bya contrapositive.a
Prove:a Fora everya integera n,a ifa n22n+7a isa even,a thena na isa odd.
Proof:a Assumea thata na isa na isa NOTa odda resultinga ina n22n+7a NOTa
beinga even.
Sincea na isa even,a n=2ka fora somea integera k.
Plugginga 2ka intoa n22n+7a givesa (2k)22(2k)+7a =a
4k24k+7a
Factora outa aa 2a thisa resultsa ina 2(2k22k+3)+1a whicha isa odd,a
meaninga ifa na isa even,a n22n+7isa odd.
Thus,a bya contrapositive,a fora everya integera n,a ifa n22n+7a isa even,a
thena na isa odd.