From 195e808eda55435ee11b264d5bd142c560bfb652 Mon Sep 17 00:00:00 2001 From: tomit4 Date: Wed, 13 May 2026 19:35:59 -0700 Subject: [PATCH] :construction: Going through 1.3 --- chapter_1/1_3/definition_1_3_4.md | 69 +++++++ ...screte_math_de_morgans_law_equivalency.png | Bin 0 -> 37015 bytes .../1_3/discrete_math_double_negation.png | Bin 0 -> 16335 bytes ...rete_math_implications_are_injunctions.png | Bin 0 -> 28269 bytes ...discrete_math_negation_and_implication.png | Bin 0 -> 38852 bytes ...discrete_math_quantifiers_and_negation.png | Bin 0 -> 19201 bytes chapter_1/1_3/double_negation.md | 5 + .../1_3/implications_are_disjunctions.md | 8 + chapter_1/1_3/investigate.md | 102 +++++++++++ chapter_1/1_3/preview_activity.md | 168 ++++++++++++++++++ chapter_1/1_3/quantifiers_and_negation.md | 5 + chapter_1/1_3/theorem_1_3_7.md | 10 ++ chapter_1/1_3/theorem_1_3_9.md | 8 + leftoff.txt | 2 +- 14 files changed, 376 insertions(+), 1 deletion(-) create mode 100644 chapter_1/1_3/definition_1_3_4.md create mode 100644 chapter_1/1_3/discrete_math_de_morgans_law_equivalency.png create mode 100644 chapter_1/1_3/discrete_math_double_negation.png create mode 100644 chapter_1/1_3/discrete_math_implications_are_injunctions.png create mode 100644 chapter_1/1_3/discrete_math_negation_and_implication.png create mode 100644 chapter_1/1_3/discrete_math_quantifiers_and_negation.png create mode 100644 chapter_1/1_3/double_negation.md create mode 100644 chapter_1/1_3/implications_are_disjunctions.md create mode 100644 chapter_1/1_3/investigate.md create mode 100644 chapter_1/1_3/preview_activity.md create mode 100644 chapter_1/1_3/quantifiers_and_negation.md create mode 100644 chapter_1/1_3/theorem_1_3_7.md create mode 100644 chapter_1/1_3/theorem_1_3_9.md diff --git a/chapter_1/1_3/definition_1_3_4.md b/chapter_1/1_3/definition_1_3_4.md new file mode 100644 index 0000000..424d24c --- /dev/null +++ b/chapter_1/1_3/definition_1_3_4.md @@ -0,0 +1,69 @@ +# 1.3.3 Logical Equivalence + +You might have noticed in Example 1.3.2 that the final column in the truth table +for $\neg P \vee Q$ is identical to the final column in the truth table for +$P \to Q$: + +| $P$ | $Q$ | $P \to Q$ | $\neg P \vee Q$ | +| --- | --- | --------- | --------------- | +| T | T | T | T | +| T | F | F | F | +| F | T | T | T | +| F | F | T | T | + +This says that no matter what $P$ and $Q$ are, the statements $\neg P \vee Q$ +and $P \to Q$ are either both true or both false. We therefore say these +statements are **logically equivalent**. + +--- + +## Definition 1.3.4 Logical Equivalence + +Two (molecular) statements $P$ and $Q$ are **logically equivalent** provided $P$ +is true precisely when $Q$ is true. That is, $P$ and $Q$ have the same truth +value under any assignment of truth values to their atomic parts. We write this +as $P \equiv Q$. + +--- + +To verify that two statements are logically equivalent, you can make a truth +table for each and check whether the columns for the two statements are +identical. + +In Section 1.2 we claimed that whenever an implication is true, so is its +contrapositive. We can now make this claim as the following theorem. + +## Theorem 1.3.5 + +_An implication is logically equivalent to its contrapositive. That is,_ + +$$ P \to Q \equiv \neg Q \to \neg P $$ + +**Proof** + +We simply examine the truth tables. + +| $P$ | $Q$ | $P \to Q$ | +| --- | --- | --------- | +| T | T | T | +| T | F | F | +| F | T | T | +| F | F | T | + +| $P$ | $Q$ | $\neg Q$ | $\neg P$ | $\neg Q \to \neg P$ | +| --- | --- | -------- | -------- | ------------------- | +| T | T | F | F | T | +| T | F | T | F | F | +| F | T | F | T | T | +| F | F | T | T | T | + +(Note that we have the truth value combinations in the same order in both +tables, so we can easily see that the final columns are identical) + +(Note that we have the truth value combinations in the same order in both +tables, so we can easily see that the final columns are identical) + +Recognizing two statements as logically equivalent can be quite helpful. +Rephrasing a mathematical statement can often lend insight into what it is +saying, or how to prove or refute it. 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