2.10 c and d

2.11 c and e

2.12 b and d

2.15 b

please show work. Thank you

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2.10 Illustrate the following equations using circuits of switches: (a) XY+XY=X (b) (X+Y)Y=XY (c) X+XZY=X+YZ (d) (A+B)C+(A+B)C=A+B (e) (X+Y)(X+Z)=X+YZ (f) X(X+Y)=X 2.11 Simplify each of the following expressions by applying one of the theorems. State the theorem used. (a) (A+B+C)(A+B+C) (b) AB(C+D)+B(C+D) (c) AB+(C+D)(AB) (d) (ABF+CD)(ABF+CEG) (e) [AB+(C+D)+EF](C+D) (f) A(B+C)(DE+F)+(DE+F) 2.12 Simplify each of the following expressions by applying one of the theorems. State the theorem used. (a) (X+YZ)+(X+YZ) (b) [W+X(Y+Z)][W+X(Y+Z)] (c) (VW+UX)(UX+Y+Z+VW) (d) (UV+WX)(UV+WX+YZ) (e) (W+X)(Y+Z)+(W+X)(Y+Z) (f) (V+U+W)[(W+X)+Y+UZ]+[(W+X)+UZ+Y] 15 Use only DeMorgan's relationships and Involution to find the complements of the following functions: (a) f(A,B,C,D)=[A+(BCD)][(AD)+B(C+A)] (b) f(A,B,C,D)=ABC+(A+B+D)(ABD+B)