**1.** Consider the four statements “if 1 = 1 then 2 = 2,” “if 1 = 1 then 2 = 3,” “if 1 = 0 then 2 = 2,” and “if 1 = 0 then 2 = 3.” The second statement is the only one that is false.

**3.** *A* and *B* are both true or they are both false.

**4.** **a.** If there are no clouds, then it is not raining. **c.** If *x* ≤ 0, then *x* ≤ 1. **e.** If *L*_{1} and *L*_{2} are parallel, then *L*_{1} and *L*_{2} don’t divide the plane into four parts. **g.** If *x* + *y* ≤ 0, then (*x* ≤ 0 or *y* ≤ 0).

**5.** **a.** It is not raining or there are clouds. **c.** *x* ≤ 1 or *x* > 0. **e.** *L*_{1} and *L*_{2} don’t divide the plane into four parts or *L*_{1} and *L*_{2} are not parallel. **g.** (*x* ≤ 0 or *y* ≤ 0) or (*x* + *y* > 0).

**6.** **a.** For example, let *x* = 2 and *y* = − 1. **c.** For example, let *x* = 3 and *y* = 2.

**7.**

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