122 FUNDAMENTALS OF SET THEORY Review Problems Problem 14.1 Show that x x = x. Problem 14.2 Show that x 0 = 0. Problem 14.3 Show that (x y) x = x. Problem 14.4 Show that x (x y) = x y. Problem 14.5 Let v S such that x v = x for all x S. Show that v = 1. Problem 14.6 Show that x y = x y. Problem 14.7 Show that 1 = 0 and 0 = 1. Problem 14.8 Show that x (x y) = x y. Problem 14.9 Show that x y = x y. Problem 14.10 Simplify [(x y z) s t] [(x y z) (s t)]. Problem 14.11 Simplify x x y. Problem 14.12 Simplify x y (x y) (y y). Problem 14.13 Simplify (a b) (a c). 14 BOOLEAN ALGEBRA Problem 14.14 Find the resulting circuit simplification of the circuit below. Problem 14.15 Simplify (a b) [a (b c)] [b (b c)]. Problem 14.16 Find the resulting circuit simplification of the circuit below. Problem 14.17 Simplify a (a b) [a (a c)]. Problem 14.18 Simplify a b b c (b c). Problem 14.19 Find the resulting circuit simplification of the circuit below. 123 124 FUNDAMENTALS OF SET THEORY Problem 14.20 Simplify (a b) (a b) b. Problem 14.21 Simplify: (A B C) C A B C D. Problem 14.22 Reduce the following logic circuit with a simple logic circuit. Relations and Functions The reader is familiar with many relations which are used in mathematics and computer science, i.e. \"is a subset of\