There is a mathematical system with 2 operators # and & and 4 different inputs/variables. The 2 operators are defined in the picture.
I need to prove that the system is boolean algebra and to find the complements of 0,a,b,and c.
Now, to prove that it's boolean algebras, I need to show that the identity, distributive, and commutative property holds.
Commutative: Chart is symmetric and since A#B = B#A, and the same for the & operator. Distributive: A#(B&C) == A#B&A#C, and the same for the & operator.
I don't know how to show the identity though and I can't figure out how to use the chart to find the complements of the 4 variables. Can someone help me out?
What $x$ makes $\;x\#y=y, y=y\#x\;$ for any $y$? that is, which row/column is identical to the title row/column? This $x$ is the $\#$-identity element.
What $x$ makes $\;x\&y=y,y=y\&x\;$ for any $y$? That is, which row/column is identical to the title row/column? This $x$ is the $\&$-identity element.
PS: Trouble finding the inverses(/complements) is not a fault on your part. You will have similar difficulty finding unicorns at the zoo.