(4) 1. Let P(x) = "x is a good singer" and Q(x) = "x has appeared on The Voice." And the u.d. for x is all people. a. Translate into English Q(Samantha) v P(Samantha) b. Translate into symbols: There exists a person who is both a good singer and who has appeared on The Voice.

The domain of discourse for this problem is a group of three people who are working on a project. To make notation easier, the people are numbered 1, 2, 3. The predicate M(x, y) indicates whether x has sent an email to y, so M(2, 3) is read "Person 2 has sent an email to person 3." The table below shows the value of the predicate M(x, y) for each (x, y) pair. The truth value in row r and column y gives the truth value for M(x, y). M | 1 2 | 3 1 TTT 2 TFT 3 TT|F Determine if the quantified statement is true or false. Justify your an- swer. (a) Væ Vy (x # y) → M(x, y)) (b) Væ3y ¬M(x, y) (c) 3r Vy M(x, y)

For "x is in the symmetric difference of A and B" give an equivalent statement that uses "x is in A"  and "x is in B".