Conditional and biconditional statements
- 1. 7.1 CONDITIONAL AND BICONDITIONAL STATEMENTS BY: JAY BASILGO MARC OVILLE DANNAH PAQUIBOT
- 2. CONDITIONAL STATEMENTS EXAMPLE: “If two distinct lines intersect, then they intersect at exactly one point.”
- 3. CONDITIONAL STATEMENTS • If – then statements •Consists of two parts: - if , hypothesis - then, conclusion
- 4. 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.
- 5. MORE EXAMPLES •If 2(x+5) = 12, then X = 1. Hypothesis: Conclusion:
- 6. MORE EXAMPLES •A quadrilateral is a polygon. •A prime number has 1 and itself as factors. •A square is a rectangle.
- 7. THE CONVERSE, INVERSE, AND CONTRAPOSITIVE OF A CONDITIONAL STATEMENT It 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 q i. Converse: If q, then p. ii. Inverse: If not p, then not q.
- 8. 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.
- 9. 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
- 10. 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.
- 11. INVERSE •To write the inverse of a conditional statement, simply negate both the hypothesis and conclusion.
- 12. 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.
- 13. CONTRAPOSITIVE •To form the contrapositive of a conditional statement, first, get its inverse. Then, interchange its hypothesis and conclusion.
- 14. EXAMPLE a. 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.
- 15. 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.
- 16. EXAMPLE a.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