Chapter 1

Describe geometrically the following sets in the \(x y\) -plane. (i) \(\\{(x, y) \mid xx^{2}\right\\}\) (v) \(\\{(x, y)|| x|+| y \mid<4\\} ;\) (vi) \(\left\\{(x, y) \mid(x-2)^{2}+(y+5)^{2} \leq 9\right\\}\) (vii) \(\\{(x, y) \mid x=0\\} ;\) (viii) \(\left\\{(x, y) \mid x^{2}-2 x y+y^{2}<0\right\\} ;\) (ix) \(\left\\{(x, y) \mid x^{2}-2 x y+y^{2}=0\right\\}\).

Prove that (i) \((A \cup B) \times C=(A \times C) \cup(B \times C)\); (ii) \((A \cap B) \times(C \cap D)=(A \times C) \cap(B \times D)\); \((\) iii \()(X \times Y)-\left(X^{\prime} \times Y^{\prime}\right)=\left[\left(X \cap X^{\prime}\right) \times\left(Y-Y^{\prime}\right)\right] \cup\left[\left(X-X^{\prime}\right) \times Y\right]\) [Hint: In each case, show that an ordered pair \((x, y)\) is in the left-hand set iff it is in the right-hand set, treating \((x, y)\) as one element of the Cartesian product.]

Is \(R\) an equivalence relation on the set \(J\) of all integers, and, if so, what are the \(R\) -classes, if (a) \(R=\\{(x, y) \mid x-y\) is divisible by a fixed \(n\\}\) (b) \(R=\\{(x, y) \mid x-y\) is \(o d d\\}\) (c) \(R=\\{(x, y) \mid x-y\) is a prime \(\\}\). \((x, y, n\) denote integers.)

Prove the distributive laws (i) \(A \cap \bigcup X_{i}=\bigcup\left(A \cap X_{i}\right)\); (ii) \(A \cup \bigcap X_{i}=\bigcap\left(A \cup X_{i}\right)\); \((\mathrm{iii})\left(\bigcap X_{i}\right)-A=\bigcap\left(X_{i}-A\right) ;\) (iv) \(\left(\bigcup X_{i}\right)-A=\bigcup\left(X_{i}-A\right)\); \((\mathrm{v}) \cap X_{i} \cup \bigcap Y_{j}=\bigcap_{i, j}\left(X_{i} \cup Y_{j}\right) ;^{4}\) (vi) \(\bigcup X_{i} \cap \bigcup Y_{j}=\bigcup_{i, j}\left(X_{i} \cap Y_{j}\right)\)

Show by examples that \(R\) may be (a) reflexive and symmetric, without being transitive; (b) reflexive and transitive without being symmetric. Does symmetry plus transitivity imply reflexivity? Give a proof or counterexample.

Prove that (i) \(\left(\bigcup A_{i}\right) \times B=\bigcup\left(A_{i} \times B\right) ;\) (ii) \(\left(\cap A_{i}\right) \times B=\bigcap\left(A_{i} \times B\right) ;\) (iii) \(\left(\bigcap_{i} A_{i}\right) \times\left(\bigcap_{j} B_{j}\right)=\bigcap_{i, j}\left(A_{i} \times B_{i}\right) ;\) (iv) \(\left(\bigcup_{i} A_{i}\right) \times\left(\bigcup_{j} B_{j}\right)=\bigcup_{i, j}\left(A_{i} \times B_{j}\right)\).