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3 Logic The Study of What’s True or False or Somewhere in Between

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3.2 Truth Tables Learn the truth tables for the five fundamental connectives Compute truth tables for compound statements (continued on next slide)

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**3.2 Truth Tables Determine when statements are logically equivalent**

State and apply DeMorgan’s laws.

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Truth Tables We want to know if a pair of similar statements such as and mean the same thing. We use truth tables to determine when compound statements are true and when they are false, and whether a pair of statements have the same meaning.

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Truth Tables

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**Truth Tables If p is true, then ~p is false.**

If p is false, then ~p is true. (example on next slide)

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Truth Tables Example:

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Truth Tables A conjunction is true only when both of its components are true. (example on next slide)

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Truth Tables Example: (solution on next slide)

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Truth Tables Example: (solution on next slide)

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Truth Tables Solution:

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**Truth Tables A disjunction is false only when both p and q are false.**

(example on next slide)

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Truth Tables Example: (solution on next slide)

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Truth Tables Solution:

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Truth Tables Because of their similar structures, there are parallels that occur in logic and set theory. Often if a result is true for set theory, then a similar result also holds for logic (and vice versa).

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Compound Statements We use truth tables to find the logical values of complex statements. Example: Compute a truth table for (solution on next 3 slides)

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Compound Statements (solution continued on next slide)

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Compound Statements (solution continued on next slide)

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Compound Statements If p is false and q is true, line 3 tells us that is false.

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Compound Statements Definition: If the final column of a truth table contains all T’s, then the statement is always true. Such a statement is called a tautology. Example: “This contestant will win or will not win.”

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Compound Statements How many lines should your truth table have?

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**Logically Equivalent Statements**

Logically equivalent statements express the same meaning. (example on next slide)

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**Logically Equivalent Statements**

Example: (solution on next slide)

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**Logically Equivalent Statements**

Solution: The two statements are logically equivalent.

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DeMorgan’s Laws (example on next slide)

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DeMorgan’s Laws Example:

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DeMorgan’s Laws

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**Truth Tables: Alternative Method**

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