Discrete mathematics Ch 1 : Propositional logic , Topic 2 : Logical operators or connectives

Question

Which of the following is the negation of x is even iff x is divisible by 2

a) (x is even or x is not divisible by 2) and (x is not even or x is divisible by 2)

b) (x is even and x is not divisible by 2) or (x is not even and x is divisible by 2)

c) x is not even iff x is not divisible by 2

d) x is even if x is divisible by 2

Answer 1

The answer is B

Let p : x is even
q : x is divisible by 2

p <=> q

lets simplify this before we negate

((p => q) and (q => p))

((¬p or q) and (¬q or p))

now negate this and we get

((p and ¬q) or (q and ¬p))

or simply say

((p and ¬q) or (¬p and q))

which is

(x is even and x is not divisible by 2) or (x is not even and x is divisible by 2)