(1) Calculate x ⊕ x, x ⊕ ¬x, x ⊕ 1, x ⊕ 0.
Solution: x ⊕ x = 0
x ⊕ ¬x = 1
x ⊕ 1 = ¬x
x ⊕ 0 = x
(2) Prove or disprove that x + (y ⊕ z) = (x + y) ⊕ (x + z)
It is not TRUE , if we assume that x is true then x + (y ⊕ z) will be true always although (x + y) ⊕ (x + z) will be FALSE always.
(3) Prove or disprove that x ⊕ (y + z) = (x ⊕ y) + (x ⊕ z)
It is not TRUE , if we assume that x is true then x ⊕ (y + z) will be equal to ¬(y + z) , that means neither of y, z, and (x ⊕ y) + (x ⊕ z) is equal to (¬y) ∨ (¬z) , it means nand of y and z. So finally we can say that they are absolutely not identical.