hw1solutions

# hw1solutions - Chapter 0 Sets and Functions(PSU Notes 1 A =...

This preview shows pages 1–2. Sign up to view the full content.

1. A = { 1 , 3 , 5 , 7 , 9 , 11 } ,B = { 3 , 6 , 9 , 12 , 15 , 18 } ,A B = { 3 , 9 } ,A - B = { 1 , 5 , 7 , 11 } . 2. (1) A B C = { 4 } . (2) A - B = { 1 , 3 } . (3) ( A B ) C = { 5 , 7 , 9 , 10 } . (4) A C B C C C = { 1 , 2 , 3 , 5 , 6 , 7 , 8 , 9 , 10 } . It is easy to see that { 4 } C = { 1 , 2 , 3 , 5 , 6 , 7 , 8 , 9 , 10 } . 4. 6. Solve ( y - 2 = x - 1 2 x + 1 = y + 2 ( y = x + 1 2 x + 1 = y + 2 2 x +1 = x +1+2 x = 2 and y = 3. 8. R = { (1 , 3) , (1 , 5) , (2 , 3) , (2 , 5) , (3 , 5) , (4 , 5) } , so dom R = { 1 , 2 , 3 , 4 } and range R = { 3 , 5 } . 10. (a) Reﬂexive: (1 , 1) , (2 , 2) , (3 , 3) , (4 , 4) , (5 , 5) R . (b) Symmetric: (1 , 2) R (2 , 1) R , (3 , 4) R (4 , 3) R . No other pairs. (c) Transitive: (1 , 2) , (2 , 1) R (1 , 1) R ; (3 , 4) , (4 , 3) R (3 , 3) R ; (2 , 1) , (1 , 2) R (2 , 2) R ; (4 , 3) , (3 , 4) R (4 , 4) R . No other pairs. Then 1 2 , 3 4 , 5 is alone, i.e. E = { 1 , 2 } ∪ { 3 , 4 } ∪ { 5 } . 12. g ( - 2) = - 7 ,g ( - 1) = 0 ,g (0) = 1 ,g (1) = 2 ,g (2) = 9. We can verify that for all x 1 ,x 2 X , x 1 6 = x 2 will imply g ( x 1 ) 6 = g ( x 2 ), so g is one-to-one, and im( g ) = {- 7 , 0 , 1 , 2 , 9 } . By solving y = x 3 + 1, we have y - 1 = x 3 , i.e. x = 3 y - 1. Then g - 1 ( x ) = 3 x - 1 (if we assume that the independent variable is always x ). 14. Since f ( a ) = b,f ( b ) = d,f ( c ) = a , and f ( d ) = c , f maps diﬀerent elements of A to diﬀerent elements in A , so f is one-to-one. Meanwhile, im( f ) = { b,d,a,c } = A , so f is onto. Its inverse function is: f - 1 ( b ) = a,f - 1 ( d ) = b,f - 1 ( a ) = c , and f - 1 ( c ) = d , i.e. f

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

hw1solutions - Chapter 0 Sets and Functions(PSU Notes 1 A =...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online