please type

1. Determine whether (pq) (pq) q.

2.Determine whether p (qr) is equivalent to (pq) r.

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

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

F(x): x is a freshman

A(y): y is an advanced course

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

Write the statement below using these predicates and any needed quantifiers:

Some freshman is taking an advanced course.

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

.Determine whether the following argument is valid:

pr

qr

qr

________

\p