Math 1090 A Homework 4 Math 1090 A Homework 4
Due November 10 at Noon.

1. Prove using the methods of the text that
 |- (a = 2) Ù(a = 3)   Þ  (2 = 3) .
2. You are given that {0,1} is the universe of discourse, Px is the predicate ``x = 0'', and Qx is the predicate ``x = 1''. Establish whether each of the following is true.
 ("x|:Px Þ Qx)
 Þ
 ((" x|:Px)Þ ("x|:Qx))
 ((" x|:Px)Þ ("x|:Qx))
 Þ
 ("x|:Px Þ Qx)
 ("x|:Px Þ Qx)
 º
 ((" x|:Px)Þ ("x|:Qx))
3. Give an example (i.e., choose a universe of discourse and examples for P, Q and y) which shows that
 (Py Þ Qy) Þ ((\$x|:Px)Þ (\$x|:Qx))
is not a theorem.

4. Fill in REASONS in the following proof that
 |- (+j | 0 £ j £ n-1 : (j+1)2 ) = (+k | 1 £ k £ n : k2 ) .

 (+j | 0 £ j £ n-1 : (j+1)2 )
 =

 REASONS

 (+j | 0 £ j £ n-1 : (+k | k = j+1:k2 ))
 =

 REASONS

 (+j,k | (0 £ j £ n-1) Ù(k = j+1):k2 )
 =

 REASONS

 (+j,k | (0 £ j £ n-1) Ù(j = k-1):k2 )
 =

 REASONS

 (+j,k | (0 £ k-1 £ n-1) Ù(j = k-1):k2 )
 =

 REASONS

 (+j,k | (1 £ k £ n) Ù(j = k-1):k2 )
 =

 Axiom, (*x,y | P:Q ) = (*y,x | P:Q )

 (+k,j | (1 £ k £ n) Ù(j = k-1):k2 )
 =

 REASONS

 (+k | 1 £ k £ n:(+j | j = k-1:k2 ))
 =

 REASONS

 (+k | 1 £ k £ n:k2 ))

File translated from TEX by TTH, version 2.60.
On 6 Nov 2000, 15:58.