DEFINITIONS AND BICONDITIONAL STATEMENTS

Definitions and Biconditional Statements :

All definitions can be interpreted "forward" and "backward". For instance, the definition of perpendicular lines means

(i) If two lines are perpendicular, then they intersect to form a right angle.

and

(ii) If two lines intersect to form a right angle, then they are perpendicular.

Conditional statements are not always written in if-then form. Another common form of a conditional statement is only-if-form.

Here is an example. We can rewrite this conditional statement in if-then form as follows :

If it is Sunday, then I am in park.

A biconditional statement is a statement that contains the phrase "if and only if". Writing biconditional statement is equivalent to writing a conditional statement and its converse.

A biconditional statement can be either true or false. To be true,both the conditional statement and its converse must be true. This means that a true biconditional statement is true both “forward” and “backward.” Alldefinitions can be written as true biconditional statements.

Definitions and Biconditional Statements - Examples

Example 1 :

Whether each statement about the diagram is true. Explain your answer using the definitions you have learned.

(i)  Points D, X and B are collinear.

(ii)  AC is perpendicular to DB.

(iii)  ∠AXB is adjacent to ∠CXD. Solution :

(i) This statement i s true. Two or more points are collinear, if they lie on the same line. The points D, X and B all lie on line DB. So thery are collinear.

(ii) This statement is true. The right angle symbol in the diagram indicates that the lines AC and BD intersect to form a right angle. So, the lines are perpendicular.

(iii) This statement is false. By definition, adjacent angles must share a common side. Because ∠AXB and ∠CXD do not share a common side, they are adjacent.

Example 2 :

Write the following biconditional statement as a conditional statement and its converse.

Biconditional Statement :

"Three lines are coplanar if and only if they lie in the same plane"

Solution :

Conditional Statement:

If three lines are coplanar, then they lie in the same plane.

Converse:

If three lines lie in the same plane, then they are coplanar.

Example 3 :

Consider the following statement :

x = 3 if and only if x² = 9

(i) Is this a biconditional statement ?

(ii) Is the statement true ?

Solution :

(i) The statement is biconditional because it contains “if and only if.”

(ii) The statement can be rewritten as the following statement and its converse.

Conditional statement :

If x = 3, then x² = 9.

Converse :

If x² = 9, then x = 3.

The first of these statements is true, but the second is false. Because, if x² = 9, then x = 3 or -3.

So, the biconditional statement is false.

Example 4 :

Each of the following statements is true. Write the converse of each statementand decide whether the converse is true or false. If the converse is true, combine it with the original statement to form a true biconditional statement. If the converse is false, state a counterexample.

(i) If two points lie in a plane, then the line containing them lies in the plane.

(ii) If a number ends in 0, then the number is divisible by 5.

Solution :

Solution (i) :

Converse :

(i) If a line containing two points lies in a plane, then the points lie in the plane.

The converse is true, as shown in the diagram. So, it can be combined with the original statement to form the true biconditional statement written below.

Biconditional statement :

Two points lie in a plane, if and only if the line containing them lies in the plane.

Solution (ii) :

Converse :

If a number is divisible by 5, then the number ends in 0. The  converse is false. As a counterexample, consider the number 15. It is divisible by 5, but it does not end in 0, as shown below.

20 ÷ 5  =  4

25 ÷ 5  =  5

30 ÷ 5  =  6 After having gone through the stuff given above, we hope that the students would have understood "Definitions and biconditional statements".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here. WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6