Geometry: Getting Started - Logical Reasoning
This is a free lesson from our course in Geometry
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...)
<h2> Geometry - Getting Started - Logical Reasoning </h2> <p> statement, negation, video, postulate, truth, value, geometry, theorem, sentence, true, false, if then, reasoning, conclusion, geometry tutorials, logical reasoning, solution, math help, practice questions, quizzes</p> <p> This mini-lesson defines and explains with the help of some examples, about a statement, truth value, negation, postulate and theorem.</p>
Other useful lessons:
Converse, Inverse, Biconditional and Contrapositive
Inductive and Deductive Reasoning
Disproving Statements
(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:
Logical Reasoning - relationship in sentences
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:
Logical Reasoning - negation of statement
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?
Logical Reasoning - mathematical reasoning problem
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 and q 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:
“Area (p) = lw”                       true
“Perimeter (q) = l+ w”             false
“Perimeter (r) = 2(l+ w)”          true
Logical Reasoning - conjunction
To do: write a complete sentence to show what the symbols represent and whether the statement is true or false.
p q Area = lw and perimeter = l+ w & T F False statement
p r 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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