Contrapositive 反證

if φ ⇒ ψ , then ¬ψ ⇒ ¬φ

$∃x$A(x)] = $∀x$[¬A(x)] ?

Assignment 7

  1. Prove or disprove the statement “All birds can fly.”

birds b, fly birds are F(b)

Assume $∀b$ ⇒ F(b) True.

$∃o$, there are some birds, such as ostrich o, it can not fly.

$∀b$[$∃o$$(b=o)$ ⇒ ¬F(b)]

so not all birds could fly, $∀b$ ⇒ F(b) False.

  1. Prove or disprove the claim (∀x, y ∈ R)[(x − y)2 > 0]. Assume (∀x, y ∈ R)[$(x − y)^2$ > 0] is True.

  2. Prove that between any two unequal rationals there is a third rational.

Let (x, y) ∈ Q rationals, (x < y), x= q/s, y=r/t

and (q, r, s, t )∈ Z integers.

then (x+y)/2 = (qt+sr)/2st ∈ Q

x < (x+y)/2 < y

  1. Explain why proving φ ⇒ ψ and (¬φ) ⇒ (¬ψ) establishes the truth of φ ⇔ ψ.

if φ ⇒ ψ , then ¬ψ ⇒ ¬φ

if (¬φ) ⇒ (¬ψ), then ψ ⇒ φ

so (φ ⇒ ψ)∧(ψ ⇒ φ)

so φ ⇔ ψ.

  1. $\sqrt{3}$ is irrational.

$\sqrt{3}$ = x/y rational, (x, y) ∈ Z

y$\sqrt{3}$ = x,

  1. converse the conditional statements.

(a)  If the Dollar falls, the Yuan will rise. F(d) ⇒ ¬F(y)

¬F(y) ⇒ F(d)

(b)  If x < y then −y < −x. (For x,y real numbers.)

(x < y) ⇒ (−y < −x)

(−y < −x) ⇒ (x < y)

(c)  If two triangles are congruent they have the same area.

 If two triangles have the same area, they are congruent.

(d)  The quadratic equation $ax^2$ + bx + c = 0 has a solution whenever $b^2$ ≥ 4ac. (Where a, b, c, x denote real numbers and a ≠ 0.)

$ax^2$ + bx + c = 0

set r, q are real numbers

(x+r)(ax+q) =0

$ax^2$ +(q+ra)x + rq = 0

b= q+ra

c = rq

b=q+ ac/q

bq-$q^2$=ac

q(q+ra)-$q^2$=ac

qra=ac

whenever $b^2$ ≥ 4ac

$(q+ra)^2$ ≥ 4qra

$q^2$ + 2qra + $(ra)^2$ ≥ 4qra

$q^2$ - 2qra + $(ra)^2$ ≥ 0

$(q-ra)^2$ ≥ 0

this is truth, so the statement is true.

  1. (e)  Let ABCD be a quadrilateral. If the opposite sides of ABCD are pairwise equal, then the

opposite angles are pairwise equal.

  1. (f)  Let ABCD be a quadrilateral. If all four sides of ABCD are equal, then all four angles are equal.

(g)  If n is not divisible by 3 then $n^2$ + 5 is divisible by 3. (For n a natural number.)

x are integers.

n=3x

( $n^2$ + 5 ) ≠ 3x

¬(n=3x) ⇒ [( $n^2$ + 5 ) = 3x] ¬ [( $n^2$ + 5 ) = 3x] ⇒ (n=3x)

($n^2$ -3x + 5 ≠ 0) ⇒ (n=3x)

($n^2$ - n + 5 ≠ 0)

$(n - 1/2)^2$ + 19/4 ≠ 0, this is true

8. Let r, s be irrationals, which is is necessarily irrational?

a. r+3

Assume r+3 = q/p rational, (q, p) ∈ Z integers.

so r= q/p- 3, it is rational also, it is converse to the condition r is irrational.

so the Assume r+3 is rational is False.

  1. 5r

the save as above.

r= q/5p

  1. r+s

r+s = q/p

r= (q-ps)/s

  1. rs

  2. √r

  3. $r^s$

  4. Let m and n be integers. Prove that:

(a) If m and n are even, then m+n is even.

set x, y integer

m= 2y

n= 2x

m+n = 2(x+y)

(b) If m and n are even, then mn is divisible by 4. m= 2x, n=2y, mn = 4xy

mn/4=xy

(c) If m and n are odd, then m+n is even.

m= 2x+1

n= 2y+1

m+n=2(x+y+1)

(d) If one of m,n is even and the other is odd, then m+n is odd. m= 2x+1

n= 2y

m+n=2(x+y)+1

(e) If one of m, n is even and the other is odd, then mn is even.

mn=2y(2x+1)

Quiz

  1. valid proof that mn is odd iff m and n are odd?

mn=2x+1

m=(2x+1)/n

  1. 16

0+4+3+3+2+0

Archimedes bath

Set cold faucet fill water rate as C , hot faucet H, both faucet will take time hr.

0.5 C = 1 H

(C+H)hr = 0.5 C = 1 H