SLIDE 3 9/24 / department of mathematics and computer science
Example: F(S)\F(T) ✓ F(S\T)
(1) var y; y 2 F(S) \ F(T) (2) y 2 F(S) ^ ¬(y 2 F(T)) (3) 9x[x 2 S : F(x) = y] (Prop. image) (4) Pick an x with x 2 S and F(x) = y (5) x 2 T (6) F(x) 2 F(T) (Prop. image) (7) y 2 F(T) (8) False (9) ¬(x 2 T) (10) x 2 S ^ ¬(x 2 T) (11) x 2 S \ T (Prop. \) (12) 9x[x 2 S \ T : F(x) = y] (Prop. image) (13) y 2 F(S \ T) (14) F(S)\F(T) ✓ F(S\T) x y A B S T F(S) F(T) Properties of ‘image’: x 2 A0 |
val
= = F(x) 2 F(A0) y 2 F(A0)
val
= = 9x[x 2 A0 : F(x) = y] NB: the other direction does not hold. Exercise: give a counterexample.
10/24 / department of mathematics and computer science
Example: F(S)\F(T) ✓ F(S\T)
Proof:
According to the property of ✓, we need to prove that all elements of F(S)\F(T) are also elements of F(S\T). Let y 2 F(S)\F(T); we need to establish that y 2 F(S\T). To this end, it suffices, by the property of image, to prove the existence of x 2 S\T such that F(x) = y. Note that, from y 2 F(S)\F(T) it follows, by the property of \, that y 2 F(S) and y 62 F(T). So, by the property of image, there exists x 2 S such that F(x) = y. It therefore remains to prove that x 2 S\T, which, according to the property
- f \ and since x 2 S, amounts to proving that x 62 T.
To prove x 62 T, we suppose that x 2 T, and derive a contradiction. From x 2 T it follows, by the property of image, that F(x) 2 F(T). Hence, since F(x) = y, we have that y 2 F(T). Since also y 62 F(T) (see above), we have thus arrived at a contradiction.
11/24 / department of mathematics and computer science
Source (1)
The source F (B0) of B0 is the set of all starting points in A of arrows with their end-point in B0.
A B F : A ! B B0 F (B0)
F (B0) def = {a 2 A | F(a) 2 B0}
Example:
F : Z ! Z with F(x) = x2 1 Then: F ({1, . . . , 10}) = {3, 2, 2, 3}
3 2 3 8 3 2
12/24 / department of mathematics and computer science
Source (2)
Property of ‘source’: x 2 F (B0)
val
= = F(x) 2 B0
Example:
Let F : A ! B and U, V ✓ B. Prove that F (U) [ F (V ) = F (U [ V ).
For all x 2 U: x 2 F (U) [ F (V )
val
= = { Property of [ } x 2 F (U) _ x 2 F (V )
val
= = { Property of source } F(x) 2 U _ F(x) 2 V
val
= = { Property of [ } F(x) 2 U [ V
val
= = { Property of source } x 2 F (U [ V )
F (U) V F (V ) U B A F : A ! B
