Simplification of boolean expressions

                                          Simplification
1. Simplify the Boolean expression
    XY′Z′+XY′Z′W+XZ′
    The above expression can be written as
    XY′Z′ (1+W) +XZ′
    =XY′Z′+XZ′ as 1+W=1
    =XZ′ (Y′+1)
    =XZ′ as Y′+1=1 
2. Simplify the Boolean expression  
    X+X′Y+Y′+(X+Y′) X′Y
    The above expression can be written as
    X+X′Y+Y′+XX′Y+Y′X′Y
    =X+X′Y+Y′ as XX′=0, and YY′=0
    =X+Y+Y′ as X+X′Y=X+Y
    =X+1 as Y+Y′=1
    =1 as X + 1=1
3. Simplify the Boolean expression
     Z(Y+Z) (X+Y+Z)
     the above expression can be written as
    (ZY+ZZ)(X+Y+Z)
    = (ZY+Z) (X+Y+Z) as ZZ=Z
    =Z(X+Y+Z) as Z+ZY=Z
    =ZX+ZY+ZZ
    =ZX+ZY+Z as ZZ=Z,
    =ZX+Z as Z+ZY=Z
    =Z as Z+ZX=Z
4. Simplify the Boolean expression
    (X+Y)(X′+Z)(Y+Z)
    The above expression can be written as
    (XX′+XZ+YX′+YZ)(Y+Z)
    =(XZ+YX′+YZ) (Y+Z) as XX′=0
    =XZY+YYX′+YYZ+XZZ+YX′Z+YZZ
    =XZY+YX′+YZ+XZ+YX′Z+YZ as YY=Y, ZZ=Z
    Rearranging the terms we get
    XZY+XZ+YX′+YX′Z+YZ as YZ+YZ=YZ
    =XZ(Y+1) +YX′+YZ (X′+1) as Y+1=1, X′+1=1
    =XZ+YX′+YZ
    Now it seems that it cannot be reduced further. But apply the following trick:
    The above expression can be written as
    XZ+YX′+YZ(X+X′) as X+X′=1
    =XZ+YX′+YZX+YZX′
    Rearranging the terms we get
    XZ+YXZ+Y X′+YX′Z
    =XZ (1+Y) +YX′ (1+Z)
    =XZ+YX′ as 1+Y=1, 1+Z=1