It's been at least 20 years since I did this, so I'm hoping some other math teachers wake up soon. As to your basic question, p implies q is a basic conditional statement. It means that you're given a statement of the "If this, then that" form. p implies q means you know that every time p is true, q is also true. So if I tell my kids: "If you argue one more time, you're grounded" and they argue, they can expect to be grounded. Some basic rules from a very rusty memory: Law of Detachment: if you know p -> q is true, and you know p is also true, you can conclude that q is true. (Mom says "if you clean your room, you can go to the movies. You clean your room. conclusion: you can go to the movies.) Law of Disjunctive Inference: If you know p V q is true, then either p is true or q is true, or both. (You know that the statement "Today either I'll do laundry or school work" is true. Then you know that either I do laundry or I do school work, or possibly both.) Law of Contrapositive: If a statement is true, so is its contrapositive. The contrapositive is what you get when you both reverse and negate the statement. So the doctor tells me that "If you diet correctly, you will lose weight." Then it must be true that "If you didn't lose weight, you didn't diet correctly"-- because if you HAD, then you would have lost the weight. DeMorgan's Law: ~(p V q) = ~P ^ ~ q....kind of like the dsitributive property for logis-- the negation hits both parts of the the disjunction along with the connector. In words: If it's not true that (I'm a mom and a wife) than either I'm not a mom or I'm not a wife. Does that help at all?