Download Different ways of saying if: implies, only if, sufficient condition

Different ways of saying if: implies, only if,
sufficient condition, necessary condition.
• If it’s an apple, it’s red.
• Being an apple is a sufficient condition for
it to be red.
• Being red is a necessary condition for it to
be an apple.
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What is the antecedent in
A sequence is convergent only if it is Cauchy?
The antecedent is “A sequence is convergent”.
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Quantifiers
What does the following statement mean?
n is even
It mean’s nothing, it doesn’t have a truth value,
because it contains a free variable. It’s not a
statement, it’s a sentence.
Given integers n and m, if n = 2m then n is
even.
We’ve quantified the sentence with the word
“given”.
For all integers n and m, if n = 2m then n is
even.
This is an example of universal quantification.
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A polynomial of odd degree has a root.
For all polynomials of odd degree, p, there exists an x ∈ R such that p(x) = 0.
Two sorts of quantifiers, universal (“for all”)
and existential (“there exists”).
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Fermat’s Last Theorem
For all n ∈ Z such that n > 2, there does not
exist (x, y, z) ∈ Z3 such that
xn + y n = z n
and x, y, z 6= 0.
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Here Z stands for the set of integers.
(R is reals, C is complexes, N is natural numbers)
Negation of quantified statements
How would you show that the following statements are false?
All apples are red
Exhibit a not red apple.
Somewhere there’s a rainbow with a pot of
gold at the end of it
Show that every rainbow does not have a pot
of gold.
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• There exists a real number y such that
y 2 = 2.
For every real number y, y 2 =
6 2.
• For all x ∈ R x2 > 0.
There exists an x such that x2 ≤ 0.
• For all x ∈ R, there exists a y such that
y 2 = x.
• For all x and for all y, there exists a z such
that z = x + y.
• For all x there exists a z such that for all
y, x + y = z.
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