Homework 13 solutions
1.
(a) Yes; if ( )(1 ) = ( )(2 ) then by definition (1 ) = (2 ). But since is an injection
this implies that (1 ) = (2 ) and now, since is injective, 1 = 2 .
(b) Yes; suppose , then there exists such that () = since is onto . But sin
Homework 14 Solutions:
1.
It suffices to show that there exists a bijection from to . (Since we know that a linear function of
nonzero slope is an invertible function from to itself) it seems reasonable to consider : ,
defined by () = , as a candidate for
ECS20
Homework 6
Exercise 1
Prove or disprove each of these statements about the floor and ceiling functions.
a) x = x for all real numbers x.
b) xy = x y for all real numbers x and y.
c)
x = x for all positive real numbers x.
Exercise 2
Show that x3 is O
ECS20
Fall 2016
Homework 3
Exercise 1
Show that this implication is a tautology, by using a truth table:
[(p q ) (p r ) (q r )] r
Exercise 2
Show that
( p q) (p r ) (q r ) is a tautology
Exercise 3
a) Let x be a real number. Show that if x2 is irrational,
ECS20
Homework 5
Exercise 1
Find these values :
a) 2.4
b) 2.4
c) 3.4
f ) 6.99
d) 3.4 e) 6.99
1 1 1
1 1
g) + h) + +
4 4 2
4 4
Exercise 2 (proof)
a) Show that the following statement is true:
If x is a real number such that x2+2=0, then x4 = -5.
b)
ECS20
Homework 2
Due October 5, 2016
Exercise 1
Construct a truth table for each of these compound propositions:
a) ( p q) ( p q)
b) ( q p ) ( p q)
c) (p q ) (p q )
Exercise 2
Construct a truth table for each of these compound propositions:
a) (p q ) (p q
ECS20
Homework 4
Proofs:
Exercise 1:
Give a direct proof, an indirect proof, and a proof by contradiction of the statement: if n
is even, then n+4 is even .
Set Theory:
Exercise 2:
Let A, B and C be sets. Show that (A-B)-C = (A-C) - (B-C)
The symmetric di