$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 1.10: Summary

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Summary

• Assertions stated in English can be translated into (and vice-versa)
• In mathematics, “or” is inclusive.
• Notation:
• $$\lnot$$ (not; means “It is not the case that”)
• $$\&$$ (and; means “Both ______ and ______”)
• $$\lor$$ (or; means “Either ______ or ______”)
• $$\Rightarrow$$ (implies; means “If ______ then ______”)
• $$\Leftrightarrow$$ (iff; means “______ if and only if ______”)
• Important definitions:
• assertion
• deduction
• valid, invalid
• tautology
• logically equivalent
• converse
• contrapositive
• Determining whether an assertion is true (for particular values of its variables)
• An implication might not be equivalent to its converse.
• Every implication is logically equivalent to its contrapositive.
• Basic laws of :
• Law of Excluded Middle
• rules of negation
• commutativity of $$\&$$, $$\lor$$, and $$\Leftrightarrow$$
• associativity of $$\&$$ and $$\lor$$
• “Theorem” is another word for “valid deduction”
• Basic theorems of :
• repeat
• introduction and elimination rules for $$\&$$, $$\lor$$, and $$\Leftrightarrow$$
• elimination rule for $$\Rightarrow$$
• proof by cases