[d] If xz = yz and z - 0 then x = y

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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2:31
AA A learn-us-east-1-prod-fleet01-xythos.C
Section 3.2 Homework (from the 5" Edition)
1. Let x, y, and z be real numbers. Prove the following:
[a] -(-x)= x
[b] (-x)y = -(xy) and (- x)- y)= xy
[c] If x = 0, then - 0 and
[d] If xz = yz and z # 0 then x = y
[e] If x 0, then x² > 0
[S] 0 <1
[g] !f x > 1, then x² > x
[h] If 0 < x < 1 then x² <1
[i] If x > 0, then > 0. If x < 0, then +< 0
[j] !f 0 < x < y, then 0 < +<!
(i) x > 0 and y > 0 or
[k] If xy > 0, then
(ii) x< 0 and v<0
[1] VNEN if 0<x< y, then x" < y"
[m] If 0 < x < y, then 0 < /x < Jy
2. Prove: If x 2 0 and×sɛ for allɛ >0 , thenx = 0
3. Prove: (x|)(y) = |xy| for all real numbers x and y.
Vx, yER
4. Prove: -| s |x – y|
5. Prove: 4- y| < c, then |x| < |r| +c
Vx, yER
6. Prove: " |x – y| < ɛ, Vɛ > 0, then x = y
Vx, yER
7. Prove: 4 x, ,X3,...x, ER, then |x, + x, +.. + x,|sx, |+ |x,| + ... + |x_|
Transcribed Image Text:2:31 AA A learn-us-east-1-prod-fleet01-xythos.C Section 3.2 Homework (from the 5" Edition) 1. Let x, y, and z be real numbers. Prove the following: [a] -(-x)= x [b] (-x)y = -(xy) and (- x)- y)= xy [c] If x = 0, then - 0 and [d] If xz = yz and z # 0 then x = y [e] If x 0, then x² > 0 [S] 0 <1 [g] !f x > 1, then x² > x [h] If 0 < x < 1 then x² <1 [i] If x > 0, then > 0. If x < 0, then +< 0 [j] !f 0 < x < y, then 0 < +<! (i) x > 0 and y > 0 or [k] If xy > 0, then (ii) x< 0 and v<0 [1] VNEN if 0<x< y, then x" < y" [m] If 0 < x < y, then 0 < /x < Jy 2. Prove: If x 2 0 and×sɛ for allɛ >0 , thenx = 0 3. Prove: (x|)(y) = |xy| for all real numbers x and y. Vx, yER 4. Prove: -| s |x – y| 5. Prove: 4- y| < c, then |x| < |r| +c Vx, yER 6. Prove: " |x – y| < ɛ, Vɛ > 0, then x = y Vx, yER 7. Prove: 4 x, ,X3,...x, ER, then |x, + x, +.. + x,|sx, |+ |x,| + ... + |x_|
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