PLEASE TYPE***

1. Determine whether (p q) (p q) q.

Ans: Both truth tables are identical:

p	q	(p q) (p q)	q
T	T	T	T
T	F	F	F
F	T	T	T
F	F	F	F

2. Determine whether p (q r) is equivalent to (p q) r.

Ans: Not equivalent. Let p, q, and r be false.

3. Using c for it is cold and r for it is rainy, write It is rainy if it is not cold in symbols.

Ans: c r.

4. In the questions below suppose the variable x represents students and the variable y represents courses, and

A(y): y is an advanced course

F(x): x is a freshman

T(x,y): x is taking y.

Write the statement using these predicates and any needed quantifiers.

Some freshman is taking an advanced course.

Ans: $x$y(F(x) A(y) T(x,y)).

5. In the questions below suppose P(x,y) is a predicate and the universe for the variables x and y is {1,2,3}. Suppose P(1,3), P(2,1), P(2,2), P(2,3), P(2,3), P(3,1), P(3,2) are true, and P(x,y) is false otherwise. Determine whether the following statement is true.

"x"y (x y (P(x,y) P(y,x)).

Ans: True

6. Determine whether the following argument is valid

p r

q r

q r

________

\ p

Ans: Invalid: p true, q true, r true