1) Let S be the set S = {a, b, c, d}. Construct one example of a binary relation on S³ that is a function and one example of a binary relation on S³ that is not a function.

2) Is the expression in each of these cases a term or a formula? Why?

a) 2x² + x 5

b) (x-1)⁴ = 0

c) (square root of 'x')

d) sin² x + cos² x > 1

e) x (x-1 < 0)

3) Let the domain of the variable x be the set of real numbers. What is the truth value of each proposition?

a) x (x 2x)

b) x (x < 2x)

c) x (x 2x)

d) x (x < 2x)

4) Consider the following statement. How would you express it as a formula of first-order logic?

For all real numbers x and for all real numbers y there is a real number z such that x + y = z.

5) Let the domain of the variable x be . How would you translate into English the expression

xy (x + y = y + x)

What property of the set of real numbers is being expressed?

6) Consider the formula

x (P(x) P(a))

and the interpretation: domain D = {1, 2, 3}, P = {(1, 2), (2, 3), (1, 3)}, a = 1. Is the formula true or false for the given interpretation? Show why.