conditional and biconditional statements
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7.1 CONDITIONAL AND BICONDITIONAL
STATEMENTSBY: JAY BASILGO
MARC OVILLEDANNAH PAQUIBOT
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CONDITIONAL STATEMENTSEXAMPLE: “If two distinct lines intersect, then they intersect at exactly one point.”
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CONDITIONAL STATEMENTS
• If – then statements•Consists of two parts: - if , hypothesis - then, conclusion
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MORE EXAMPLES
•If two points lie in a plane, then the line containing them lies in the plane.
Hypothesis: Two points lie in a plane. Conclusion: The line containing them lies in the plane.
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MORE EXAMPLES
•If 2(x+5) = 12, then X = 1. Hypothesis: Conclusion:
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MORE EXAMPLES
•A quadrilateral is a polygon. •A prime number has 1 and itself as factors.•A square is a rectangle.
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THE CONVERSE, INVERSE, AND CONTRAPOSITIVE OF A CONDITIONAL
STATEMENTIt is said that if a statement is true, its contrapositive is also true.Moreover, if the converse is true, its inverse is also true.
Consider the statement: if p, then qi. Converse: If q, then p.
ii. Inverse: If not p, then not q.iii. Contrapositive: If not q, then p.
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CONVERSE•To write the converse of a conditional statement, simply interchange the hypothesis and the conclusion. That is, the then part becomes the if part.•Note that converse of a conditional statement is not always a true statement.
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EXAMPLE •a. If m<A= 9, then m<A is a Rigth angle. Converse: If m<A is a right angle, then m<A = 90
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EXAMPLE •B. The intersection of two distinct planes is a line.
If-then: If two distinct planes intersect, then their intersection is a line. Converse: If the intersection of two figures is a line, then the figure are two distinct planes.
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INVERSE
•To write the inverse of a conditional statement, simply negate both the hypothesis and conclusion.
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EXAMPLE• If m<A<90, then <A is an acute angle. INVERSE: If m<A is not 90, then <A is not an acute angle.• If two distinct lines intersect, then they intersect at one
point. INVERSE: if two distinct lines do not intersect, then they do not intersect at a point.
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CONTRAPOSITIVE•To form the contrapositive of a conditional statement, first, get its inverse. Then, interchange its hypothesis and conclusion.
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EXAMPLEa. If m<A +m<B= 90, then <A and <B are
complementary. Inverse: if m<A + m<B is not equal to 90, then <A and <B are complementary. Contrapositive: If <A and <B are not complementary, then m<A + B is not equal to 90.
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BICONDITIONAL STATEMENT• If a conditional statement and its converse are both true. Then they can be joined together into a single statement called biconditional statement. Then this is done by using the words if and only if.
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EXAMPLEa.If a+7= 12, then a = 5. Conditional statement: if a+7 = 12, then a = 5 Converse statement : If a = 5, then a + 7 = 12 *Both the conditional statement and its converse are true statements. Hence, the biconditional statement is
a+7 = 12 iff a =5