Mate Pa Subir

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PRACTICA DIRIGIDA Nº 01 1)demostrar que: a) Si x∈-3,5→0<x+37<57 Solución 0<x+3<5 0<x+37<57→l.q.q.d. b)

Si x∈-3,5→1x+5∈0,1 solución -3<x<5 2<x+5<10 110<1x+5<12 0<0.2<1x+5<0.5<1 1x+5∈0,1 l.q.q.d.

c)

Si x∈-1,5→32x+5∈0,1 Solución -1<x<5 3<2x+5<15 15<32x+5<1 0<0.2<32x+5<1 0<32x+5<1 l.q.q.d.

d)

Si

x∈2,4→12x+3∈111,17 2<x<4 4<2x<8 7<2x+3<11 111<12x+3<17 12x+3∈111,17 l.q.q.d.

2.determinarel menor M y mayor m tal que si xє [-2,3] entonces X≤X+5X+7≤M solucion X+5X+7=1-2X+7 X∈-2,3→-2≤X≤3 -25≤-2X+7≤-15 35≤1-2X+7≤45

m=35≤X+5X+7≤45=M ∴m=35

M=45

6.- Sea. A = { xє Ν / 3 ≤ x ≤10 } N/x>7} C = { 7, 8, 9, 10 } . C U (B

,

B={xє

Hallar :( A – B ) U

-Ac )

SOLUCION A = { xє N / 3 ≤ x ≤ 10 } N/x>7}

B={xє

A = { 3, 4, 5, 6, 7, 8, 9, 10 } 9, 10, 11, ... }

B = { 8,

C = { 7, 8, 9 , 10} Hallar :

(A–B) U

C U (B

{ 3, 4, 5, 6, 7 }

∩Ac

)

U { 7, 8, 9, 10 } U { ( B

∩ Ac ) }

{ 3, 4, 5, 6. 7, 8, 9, 10 }

U



{3 ,4, 5, 6, 7, 8, 9, 10 } Rpta.

11) Dados los conjuntos : }

A = { x ∈R /

1-x22 ∈ < -2 ;4 ]

B={x /

R / 2x2+ 3x+1 <0 } ,



C={x

x2 >2 → x2 < -2 }

Hallar : ( A- B )

U

(B

∩C)

SOLUCION : A={x

∈R

/ 1-x22

∈ <-2 ;4 ] }

-2 < 1-x22 ≤4 -4 <1-x2 ≤ 8 -5 < -x2 ≤ 7

por (-1)

-7 ≤ x2 <5 x2 ≥ -7 ⋀

x2 <5



x2 = 7 ⋁ x2> -7

-5 <x < 5

x ∈ ∅ ⋁ x ∈R

A.

x ∈ <-5 ; 5 >

B ={

x ∈R /

2x2+3x+1 <0 }

2x2+3x+1 <0 2x+1x+1<0

Por puntos críticos: x ∈ < -1 ; -12 >

C = {

x ∈R / x2 >2 → x2 < -2 }



x>2 -2<x<2 x ∈ <- 2



;

x∈∅

2 >

Hallar : ( A

–B)U(B∩C)

< -5 ; -1 ] U [-12;5 > U < -1 ; -12 > x ∈ < -5 ; 5>

Rpta.

x∈∅

x= -12 ; x= -1



R

12)

Dado los conjuntos : A = { x ∈R x2-3 ∈[ -3 ;6 > -3 ≤ x2- 3<6 0 ≤ x2 <9



x2≥0

x ∈R

x ∈ < -3 ;3 >

A= B=

x ∈ <-3 ;3 >

{ x ∈R / 3-2x ∈[ -4 ;2 ] -4 ≤3-2x ≤ -1

por menos 1

-7 ≤ -2x ≤ -1

entre 2

1 ≤2x ≤7 12 ≤x ≤72

B=

x ∈[ 12 ;72 ]

C = {

x ∈ R / x ∈A →x ∈B } x ∈ Ac



x∈B

< -∞ ;-3 ] U [ 3 ; +∞ > ⋁ [ 12;72 ]

C={ D = {x ∈R / x∈( A ⋀

[

x ∈ < -∞ ; -3 ] U [12 ; + ∞>

x ∈ A ↔ x ∈C }

C) ⋁

x ∈ ( Ac ⋀ Cc

12;3 > U ( ( < -∞ ;-3 ] U [ 3 : +∞ >) ⋂

[12 ;3 > U ∅ x ∈R / x∈[ 12;3>

Rpta.

) <-3 ;12 ]

)

13. Dados los conjuntos A={xєR/ 22x+3 ∈ <14 , 2]} ; B ={ xєR/x2X-2 ≥X+6}; C={ xєR/3x2-4X-6≤X+6} Hallar el conjunto M=

{

xєR/x∈(C-A)→X∈B}

SOLUCION A= 1/4<22x+3≤2 1/8<12x+3≤1 1≤2x+3<8 -2≤2x<5 -1≤x<52 x∈[-1,5/2> B=

x2X-2 ≥X+6

x2≥X+6(X+2) x2≥x2-2X+6X-12

12≥4X 3≥X….→X≤3 X∈<-∞,3] C=

3x2-4X-6≤X+6

3x2-4≤x2+6x-6x-36 2x2≤-32 x2≤-16 Ø=

vacio.

EQUIVALENCIA b) x∈A→X∈B ↔X∈Ac∨X∈B C-A…….. (Ø= vacio) - ( x∈[-1,5/2>) …………= Ø= vacio……A X∈<-∞,3]……B

Ac=<-∞,-1>∨[5/2,∞> X∈B=<-∞,3]

Luego: {<-∞,-1>∨[5/2,∞>} ∨ {<-∞,3]}

……….. rpt.

{

x∈ R;<-∞,∞> }

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