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Bellringer

Explore logic tables, postulates, and diagrams in mathematics, including AND and OR logic tables, new and old postulates, and the use of diagrams. Includes solutions.

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Bellringer

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  1. Bellringer • Block 2: Quizlets VENN and TRT. You have 5 minutes. • Blocks 1 & 3: • Write a logic table that you think describes p and q both being true at the same time (“AND”). Use the symbol ‘&’. • Write a logic table that you think describes at least one of them (p & q) being true (“OR”). Use the symbol ‘|’. • Write a truth table that finds the values for p -> q and p & ~q. Do you see a relationship? Are they equivalent? Solutions are on the next slide.

  2. Bellringer Solutions p -> q and p & ~q aren’t equivalent – they’re opposites. Whenever one is true, the other is false. This makes sense: p & ~q means that the hypothesis is true, but the conclusion is false. That’s the only time that p -> q is false.

  3. Use Postulates and Diagrams Section 2.4

  4. Objectives & Announcements • Add new postulates to our repertoire • Recognize the use of postulates in diagrams • Diagram postulates HW for next time: page 134-136, #1-13. Test on 2.1-2.4 next class! We will review before the test.

  5. Old Postulates • From Chapter 1: • Postulate 1: Ruler Postulate • Postulate 2: Segment Addition Postulate A B C AB + BC = AC • Postulate 3: Protractor Postulate • Postulate 4: Angle Addition Postulate mAVB + mBVC = mAVC A B V C

  6. New Postulates (Point, Line, & Plane) • #5: Through every two points, there is exactly one line. • #6: Every line contains at least two points. • #7: If two lines intersect, their intersection is exactly one point. • #8: Through any three noncollinear points, there exists exactly one plane. • #9: A plane contains at least three noncollinear points. • #10: If two points lie in a plane, then the line connecting them lies in the plane. • #11: If two planes intersect, then their intersection is a line.

  7. Postulate #5: Through every two points, there is exactly one line. • As with all postulates, this should be obvious. • It lets us use the notation AB to refer to the line through points A and B. Without this postulate, we wouldn’t know that there is such a line – and there could be more than one. • Example of more than one: • Look at the North and South poles of a globe. • If we allow lines to be drawn on the sphere, there are many lines going from the North Pole to the South Pole (lines of longitude). • Since we draw lines on planes instead of spheres, this does not happen.

  8. Postulate #6: Every line contains at least two points. • Every line actually contains an INFINITE number of points. • This postulate mentions only two because it’s a less strict requirement. • In Math, we try to keep the postulates as non-restrictive as we can.

  9. Postulate #7: If two lines intersect, their intersection is exactly one point. P • To see why we need this, remember the globe: • Any “line” going through the north pole would also go through the south pole. • These are lines of longitude. • All such lines would intersect in two points instead of one! • We are drawing our lines on planes, so that cannot happen.

  10. Postulate #8: Through any three noncollinear points, there exists exactly one plane. • Remember the triangle we created with string the first week of school? That triangle is part of the plane we’re talking about. • If the three points were collinear, we could have many planes through them all – in fact, an infinite number. • This is a lot like Postulate #5 (through any two points there is exactly one line).

  11. Postulate #9: A plane contains at least three noncollinear points. • There are actually infinite points – we’re just trying to be non-restrictive again. (Three is a weaker requirement). • The three points form a triangle in the plane. • This is like Postulate #6 (a line contains at least two points).

  12. Postulate #10: If two points lie in a plane, then the line connecting them lies in the plane. • We know that there is such a line because of Postulate #5. • This is an example of why Postulate 5 is important.

  13. Postulate #11: If two planes intersect, then their intersection is a line. • There was a question about this (along with a diagram) on the Chapter 1 test. • Example: • The floor of the classroom intersects with the front wall of the classroom. • Their intersection is the line along the bottom of that wall.

  14. Solutions are on the next slide. No peeking!

  15. Solutions are on the next slide.

  16. Diagrams Lie! • Reminder: Diagrams are often misleading. Here are some examples.

  17. Perpendicular Figures • This is an example of where symbols such as the red right angle marker are important. • Without them, we would not be able to assume that line t really is perpendicular to the plane.

  18. A 3-D Diagram The solution is on the next slide.

  19. A 3-D Diagram • A, B, and F are collinear, since line AF is shown and B is on it in the diagram. • E, B, and D are collinear don’t have such a line shown. We can’t assume. • Segment AB is shown with a perpendicular mark, so we know that it is  plane S. • Segment CD doesn’t have such a mark; we can’t assume that it’s perpendicular to plane T. • The diagram clearly shows that lines AF and BC intersect at point B, so we know that is true.

  20. Classwork • You have a handout with the pages needed for this assignment. • Do #3-8, 11-13, 14-23, 26, 29, 31, 32, 39, 42, 45.

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