(8 points) Problem 2.1.1
Let A be any set. What are the direct products ∅ × A and A × ∅? If x is any thing, what are the direct products A × {x} and {x} × A? Justify your answers.
(10 points) Problem 2.1.5
Let n be a natural and let I(x) be a unary relation on the set {0, . . . , n − 1}. Let w be the binary string of length n that has 1 in position x whenever I(x) is true and 0 in position x when I(x) is false. (As in Java, we consider the positions of the letters in the string to be numbered starting from 0.) What is the string corresponding to the predicate I(x) meaning “x is an even number” in the case where n = 5? The case where n = 8? If w is an arbitrary string and I(x) the corresponding unary predicate, describe the set corresponding to the predicate in terms of w.
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