Question

prove that xy = x (x+y) using truth table and boolean identities.

Answer 1

w, in this case we have 2 variables, boolean variable x and boolean variable y.

In Boolean Algebra variables can only take 2 values: TRUE or FALSE, or T or F, for short.

So the n-tuple (x, y) can only take 4 values:

(F, F), (F, T), (T, F) and (T, T)

Test the value on the right for each case:

(F, F) : x=F, y = F → xy = F&F = F ; xy’ = F&T = F → xy+xy’ = F+F = F

(F, T) : x=F, y = T → xy = F&T = F ; xy’ = F&F = F → xy+xy’ = F+F = F

(T, F) : x=T, y = F → xy = T&F = F ; xy’ = T&T = T → xy+xy’ = F+T = F or T = T

(T, T) : x=T, y = T → xy = T&T = T ; xy’ = T&F = F → xy+xy’ = T+F = T

You can see that the cases where the expression on the right is true are (T, F) and (T, T) … that is, the cases when x is TRUE. If you made the 2 variables truth table for the expression “x” you would get the same… so the part in the left is the same as the part in the right, no matter the boolean values of x and y.

I tested ALL the possible values, there aren’t more possible values and in all the cases the expressions in the left and in the right are the same… so, the expressions are ALWAYS the same.