1. Suppose A = {1, 2}, B = {a, b, c} and C = {3, 4}. What is the set A × B? What

Question

1. Suppose A = {1, 2}, B = {a, b, c} and C = {3, 4}. What is the set A × B? What is the set A × A × C? What is the set A × (A × C)?

2. Note that a function f : R R is 1-1, when its graph G = {(x, f (x)) R 2 | x R} satisfies the following horizontal line test: Every horizontal line in R 2 intersects G in at most one point. Also, note that f : R R is onto, if it satisfies: Every horizontal line in R 2 intersects G in at least one point. Write down functions f : R R satisfying each of the following conditions:

a. f is 1-1 but not onto.

b. f is onto but not 1-1.

c. f is both 1-1 and onto.

d. f is neither 1-1 nor onto.

3.Let A = {1, 2, 3, 4, 5}. Then = {{1, 2}, {3, 4, 5}} is a partition of A. Give three different examples 1, 2, 3 of partitions of A that are different from and where 1 contains the subset {1, 2, 3} as one of its elements, but no subset in 2 or in 3 has three elements.

Explanation / Answer

1. A x B is the set of all possible combinations of A and B.

A = {1, 2}, B = {a, b, c}

A X B = {(1, a), (1,b), (1,c), (2,a), (2,b), (2,c)}

A × A = { (1,1), (2,2)}

C = {3, 4}

A × A × C = {(1,1,3), (2,2,3), (1,1, 4), (2,2, 4)}

(A × C) = {(1,3), (1,4), (2,3), (2,4)}

A × (A × C) = {(1,1,3),(1,1,4), (1,2,3), (1,2,4), (2,1,3), (2,1,4), (2,2,3), (2,2,4)}

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