(b) P→r P Show that each argument is not valid by finding a truth assignment of the variables that makes the hypotheses true, but the conclusion false. (Hint: fill out a truth table if you have trouble finding an assignment.)

Y(Z+U)+Y'(W+X) using distributive laws, consensus , and theorem of factoring  obtain the pos expressions

[3] Simplify the following expressions to (1) sum-of-products and (2) products-of- sums: (d) ABC + ABD + BCD

EXAMPLE: The sum of any two even integers is even. Answer: Proof. (direct) Suppose x and y are even integers. By definition of even, let x = 2k and y = 2j for some integers j and k. Then, x+y=2k+2j = 2(k + j) Note that k+j is an integer because it is the sum of integers. Therefore, by definition of even, x + y is even.