Question Based On Logic (Mathematics)

Question Based On Logic (Mathematics)
Sometimes we say that an operator ⊗ is “left-distributive” over an operator ⊕ if we have x ⊗
(y ⊕ z) = (x ⊗ y) ⊕ (x ⊗ z). Show whether (or not) ∧ is left-distributive over →.
Proof
Let the ∧ and ⊕ be AND and additive operations respectively on a set S
then ∧ is said to be left-distributive for all x, y, z ∈ S if
x ∧ (y ⊕ z) = (x ∧ y) ⊕ (x ∧ z) .... Equation 1
For all x, y, z ∈ S and x, y, z ∈R
applying ∧ to the statement => x ∧ (y ⊕ z)= (x ∧ y) ⊕ (x ∧ z),
the distributive law implies that
for all x, y, z ∈ S and x, y, z ∈R
x ∧ (y ⊕ z)= ( x ∧ y) ⊕ (x ∧ z) .... Equation 2
the results obtained from the left hand side of equation 2 is equal to the
right hand side of equation 1.
Therefore ∧, ⊕ is left-distributive for all x, y, z ∈ S: ∧: R⊕ R→R