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Science, Technology, Engineering and Mathematics
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Notes on Logic
These notes are useful to people just starting out in formal mathematics and logic. You might want to have them to hand whilst thinking about problems such as
Mind Your Ps and Qs
At a basic level proof is based on the concepts of:
IF (something is true) THEN (something else is true)
There are three versions of this of this
(p is true) ONLY IF (q is true) [we say q is NECESSARY for p]
(note: in this case p cannot be true if q is false)
(p is true) IF (q is true) [we say q is SUFFICIENT for p]
(note: in this case p might be true if q is false)
(p is true) IF AND ONLY IF (q is true) [we say q is NECESSARY AND SUFFICIENT for p]
(note: in this case p and q are either both true or both false. They are logically equivalent)
Statements in mathematical logic are sentences which are either true or false.
We can link statements using AND, OR and NOT
(p AND q) is true if and only if BOTH p and q are true
(p OR q) is true if and only if at least one of p or q is true
NOT(p) is true if and only if p is false.
$\Leftarrow, \Rightarrow, \Leftrightarrow$
These show the direction of the IF ... THEN logic as follows.
$p\Rightarrow q$ read as
p implies q
if p is true then q is also true
$p\Leftarrow q$ read as
p is implied by q
if q is true then p is also true
$p\Leftrightarrow q$ means $p\Rightarrow q$ AND $p\Leftarrow q$.
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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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