The principle of
logic
allow you to determine if a statement
is true, false,
or uncertain in line with the truth of related statements. You can solve the
problems and draw conclusions by reasoning in Geometry from what you know to
be true. The reasoning will be based on the ways in which you’ll put
sentences together.
When it can be determined that a statement is true or that is false, that statement
is said to have a truth value. A declarative statement which may be either true
or false is called mathematical sentence. For example, “The Brooklyn bridge is in
California” is a false mathematical sentence, and “Congruent angles are angles that
have the same measure” is a true mathematical sentence.
(More text below video...)
(Continued from above)
In relation with truth values, true and false are indicated by T and F.
E.g. the statement '2 + 3 = 5' has truth value T. The relationship of
different kind of sentences can be understood in the diagram shown below:
Notice that the negation of a statement always has the opposite truth value of the
original statement and can be formed by adding word not to the given statement.
E.g. the measure of an obtuse angle is ‘not’ greater than 90º. Further,
while postulate is a proposition that is accepted as true in order to provide a
basis for logical reasoning, a theorem is a statement that has been proven, or can
be proven, from the postulates. Further understanding and learning in this lesson
will help you build skills in enhancing logical approach for geometry involved applications.
To move forward learn about the symbols in logic: Notice that a single letter is
used to represent a single complete thought. To show a negation of simple statement,
symbol ~ can be placed before the letter for given statement. Thus ~p word
represent as ‘not’ p. For example:
Remember: A statement and its negation have opposite truth values. What should be
the approach now! Generally this is what you need to understand on mathematical
reasoning problems: quite often, you may find formal mathematical reasoning in the
given problems or questions. Importantly you must note the words used and their
meaning. This might seem complicated here in text, but once you have instructor
explain it using video and in own handwriting with examples, will be easy for you
to understand. E.g. Given the triangle shown below, which of the following must
be true?
To solve it, you may take the following approach: Notice that it involves p
and r, so attempt substituting a number for x. Now think about
what is given and what you know:
• assume x equals 4
• then the side opposite m75, is 4 + r
units long
• the side opposite m80, is 4 + p units
long
• since 75 < 80, obviously 4 + r < 4 + p
• thus, r < p or p > r
• p > r is true and the correct answer.
Additional commonly used terms and concept: Conjunction a compound sentence formed by using the word and
to
combine two simple sentences. E.g. when two simple sentences, p and
q, are joined
to form a compound sentence, the conjunction is symbolically expressed as pq.
Note: The conjunction p andq is true only when both parts are true i.e. p must be true, and q must be true. Opposite to this, when both parts are false, then p and q is false. For example:
figure below is a rectangle with length l and width w. For each statement, the truth value is noted below:
To do: write a complete sentence to show what the symbols represent and whether
the statement is true or false.
Answers:
pq Area = lw and perimeter = l+ w & T
F False statement
pr Area = lw and perimeter = 2(l+ w) & T
T True statement, and so on ...
Disjunction is a compound sentence formed by combining two simple sentences using
the word or. E.g. when two simple sentences, p and q, are
joined to form a compound sentence, the disjunction is symbolically expressed as
p V q. Note: The disjunction p or q is true when any part of the compound sentence is true, say p is true, or q is true, or both p and q are true.
For example: Given the following two statements, p and q, write their disjunction
and conjunction.” if p, then q.”
Let p represent “Two lines form a right angle.” AND Let q represent “Two lines are
perpendicular”.
Notice that the complete sentence; what the symbols represent…disjunction will be:
“Two lines form a right angle or two lines are perpendicular.” And conjunction will
be: “Two lines form a right angle and two lines are perpendicular.
Remember: The only way for a disjunction to be a false statement is if both half
statements are false. A disjunction is true if both statements are true or either
statement is true. E.g. “The clock is slow or the Time is correct”, is a false
statement only if both parts are false.
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