Beginning Notes
- When p→q, q→p is not necessarily true
- If a statement and its converse are both true, the statement is known as biconditional
Biconditional: p→q and q→p
^ ^
both true
Equivalent: {p<-->q} p
if and only if q (if and only if=iff)
<---> = iff (biconditional)
The concept of 'biconditionality' can be further explained using the comparison of "word=definition" because a word 'equals' its definition, and the definition also 'equals' the word. For a statement to be considered biconditional, p→q and q→p must both be true, or in a sense, equal.
Truth Tables
We then proceeded to discuss Truth Tables, and the basic outline of this concept is shown below on this worksheet:
These tables help you understand the many comparisons between the values 'p' and 'q', and use basic logic and venn diagram reasoning to make it simpler. The following is a brief explanation of the table logic using diagrams and basic logic.
1st Table: (Not) If p is true, then not p (~p) must be false. If p is false, then not p (~p) must be true.
2nd Table: (And) If p and q are both true, then a common value of p and q is possible (true). If p is true and q is false, then a common value of p and q is not possible (false). If p is false and q is true, then a common value of p and q is impossible (false). If both p and q are false, then obviously there are no common values (false).
3rd Table: (Or-inclusive) I will use the venn diagram to explain this one. Imagine that the p circle and the q circle are shaded, as well as the intersecting portion (because it is inclusive). The shaded portion will represent true. If a point is in p and q, it is in the middle, so therefore it is true. If a point is in p but not q, it is also in the shaded region, so true. If a point is in q but not p, it can be valid as well. If a point is not in p or q, it is obviously not in the shaded region, and therefore false.
4th Table: (Conditional/implies) I will also use the above diagram to explain this table. Becaue p is inside of q, a point in p will be in both, so this is true. A point cannot be in p but not q, so this is false. A point can be in q but not p, so this is true. A point can be outside of both rings, so this is true as well.
5th Table: (Biconditional) The simple diagram is also very helpful for this table. A biconditional diagram is drawn as one circle with both letters in the same space, so this table should be easy. A point can be in p and q because they are the same circle, so this is obviously true. A point cannot be in p but not q, or q but not p, also because they are the same circle. A point can be outside of this circle of equivalence, and therefore not be in either one, so this statement is true.
This basically sums up what we learned today in Honors Geometry. Thank you for reading!
Sincerely,
Julia Wilkins
