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) (a = 3)     (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 Qx)
    ((" x|:Px) ("x|:Qx))
    ((" x|:Px) ("x|:Qx))
    ("x|:Px Qx)
    ("x|:Px Qx)
    ((" x|:Px) ("x|:Qx))
  3. Give an example (i.e., choose a universe of discourse and examples for P, Q and y) which shows that
    (Py Qy) (($x|:Px) ($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) (k = j+1):k2 )
    =

    REASONS

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

    REASONS

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

    REASONS

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

    Axiom, (*x,y | P:Q ) = (*y,x | P:Q )

    (+k,j | (1 k n) (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.