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3A Logic Name__________________________________ I. Conditional Statements (p→q) If you are driving 80 mph, then you are breaking the law. 1. Converse (q→p) If you are breaking the law, then you are driving 80 mph. 2. Inverse (~p→~q) If you are not driving 80 mph, then you are not breaking the law. 3. Contrapositive (~q→~p) If you are not breaking the law, then you are not driving 80 mph. 4. Biconditional (p⇔q) You are driving 80 mph if and only if you are breaking the law. II. Using a conditional statement 5. If two angles form a linear pair, then they are _____. True False If two angles form a linear pair, then they are supplementary. 6. Converse (q→p) True False If two angles are supplementary, then they form a linear pair. 7. Inverse (~p→~q) True False If two angles do not form a linear pair, then they are not supplementary. 8. Contrapositive (~q→~p) True False If two angles are not supplementary, then they do not form a linear pair. True False 9. Biconditional (p⇔q) Two angles form a linear pair if and only if they are supplementary. III. Conditional Cards Select a statement for the conditional statement. Then select its converse, inverse, and contrapositve. Select a different card for the conditional statement. Then select its converse, inverse, and contrapositve. IV. Segment Addition Postulate 10. What happens when B is between A and C? AB+BC=AC 11. Given: B is between A and C. Suppose AB=x+1, BC=x+4, and AC=3x–2. What is the value of x? What are the values of AB, BC, and AC? x+1+x+4=3x–2 2x+5=3x–2 7=x AB=7+1=8 BC=7+4=11 AC=3(7)–2=19 V. Angle Addition Postulate 12. m∠ABC+m∠CBD=m∠_____ m∠ABC+m∠CBD=m∠ABD 13. Suppose m∠ABD=5x+1, m∠ABC=x+10, and m∠CBD=2x–1. What is the value of x? What are the values of m∠ABD, m∠ABC, and m∠CBD? 5x+1=x+10+2x–1 m∠ABD=5(4)+1=21 5x+1=3x+9 m∠ABC=4+10=14 2x=8 m∠CBD=2(4)–1=7 x=4 VI. Logical Conclusions 14. If ∠1 and ∠2 form a linear pair, and ∠1 is obtuse, then …. ∠2 is acute. sur
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15. If two lines, AB and !CD , intersect at P, then ∠APC and ∠BPD …. are vertical angles, and they are congruent. 16. If a line bisects !AB at point M to form segments !AM and MB , then …. AM=MB, and M is the midpoint of !AB . 17. If the sum of two angles of a triangle is less than 90°, then …. the triangle is obtuse. VII. Practice 18-­‐22: Given the following conditional statement: If two angles form a linear pair, then they are adjacent. 18. Write the converse. If they are adjacent, then two angles form a linear pair. 19. Write the inverse. If two angles do not form a linear pair, then they are not adjacent. 20. Write the contrapositive. If they are not adjacent, then two angles do not form a linear pair. 21. Write the conditional statement as a biconditional. If they form a linear pair if and only if two angles are adjacent. 22. Consider the truth values of the original, converse, inverse, and contrapositive. Which, if any, are true? Which, if any, are false? Draw a counterexample for any false statements. Conditional – T; Converse – F; Inverse – F; Contrapositive – T; Biconditional – F 23. Claim: If n is even, then n+1 is divisible by 3. Is it true? Or can you find a counterexample? False. n = 4 24-­‐25: Valid or Invalid Arguments. For each of the following examples, decide if the argument is valid or invalid. 24. If someone buys a new Lamborghini, she or he will pay over $200,000. Marie does not buy a new Lamborghini. Therefore Marie does not pay over $200,000 for her new car. 25. In China, job applicants do not ask how much they will be paid when they are hired. When Jin Tai was hired, he asked his employer how much he would be paid. Jin Tai must have been hired outside of China. 26. The following statement is true: If a figure is a triangle, then it is a polygon. Which other statement must also be true? A. If a figure is not a triangle, then it is not a polygon. B. If a figure is not a polygon, then it is a triangle. C. If a figure is a polygon, then it is a triangle. D. If a figure is not a polygon, then it is not a triangle. 3A Proof Name______________________________________ I. The "Letter Game" Undefined terms: Letters M, I, and U Definition: x means any string of I's and U's Postulates: 1. If a string of letters ends in I, you may add U at the end. 2. If you have Mx, then you may add x to get Mxx. 3. If three I's occur, that is III, then you may substitute U in their place. 4. If UU occurs, you drop it. Example. Given: MI 1. Given: MIII 2. Given: MIIIUUIIIII Prove: MIIU Prove: M Prove: MIIU Statements Reasons Statements Reasons Statements Reasons 1. MI 1. GIVEN 1. MIII 1. GIVEN 1. MIIIUUIIIII 1. GIVEN 2. MII 2. P2 2. MU 2. P3 2. MIIIIIIIII 2. P4 3. MIIU 3. P1 3. MUU 3. P2 3. MIIIIIU 3. P3 4. M 4. P4 4. MIIUU 4. P3 5. MII 5. P4 6. MIIU 6. P1 3. Given: MI 4. Given: MI 5. Given: MIIIUII Prove: MUI Prove: MIUIU Prove: MIIUIIU Statements Reasons Statements Reasons Statements Reasons 1. MI 1. GIVEN 1. MI 1. GIVEN 1. MIIIUII 1. GIVEN 2. MII 2. P2 2. MIU 2. P1 2. MUUII 2. P3 3. MIIII 3. P2 3. MIUIU 3. P2 3. MII 3. P4 4. MUI 4. P3 4. MIIU 4. P1 5. MIIUIIU 5. P2 II. Jumbled Proofs uur
6. Given: !AC bisects ∠DAB, ∠1≅∠3 Prove: ∠1≅∠2 Jumbled Statements J. ∠1≅∠3 Jumbled Reasons W. Definition of angle bisector Statements Reasons M X K. ∠1≅∠2 X. Given J Y Y. Given L W K Z L. ∠2≅∠3 uur
Z. Substitution Property M. !AC bisects ∠DAB 7. Given: ∠A is a right angle, ∠B is a right angle Prove: ∠A≅∠B Jumbled Statements J. m∠A= m∠B Jumbled Reasons W. Definition congruent angles Statements Reasons M Y K. m∠A=90°, m∠B=90° X. Definition of right angle K X L. ∠A≅∠B Y. Given J Z M. ∠A is a right angle, ∠B is a right angle Z. Substitution Property L W III. Proofs 8. Given: !AC ≅!BD Prove: !AB ≅!CD Statements Reasons 1. Given 1. !AC ≅!BD 2. AC=BD 3. AB+BC=AB; BC+CD+BD 4. AB+BC=BC+CD 5. AB=CD 2. Definition of congruent segments 3. Angle Addition Postulate 4. Substitution Property of Equality 5. Subtraction Property of Equality 6. Definition of congruent segment 6. !AC ≅!BD IV. Prove It! Undefined terms: The letters A, B, C, and D. Property 1: Any two adjacent letters can change place with one another. This is similar to the Commutative Property. Example: CAD→ACD. Property 2: If a string of letters begins and ends with the same letter, all of the letters between them may be replaced with the letter B. This is similar to the Substitution Property. Example: CCAABC→CBC. Property 3: If the first two letters of a string are the same, they may be replaced with the letter A. This is similar to the Substitution Property. Example: CCAD→AAD. Example. DCCADA→AB Statements Reasons 1. DCCADA 1. GIVEN 2. DCCAAD 2. P1 3. DBD 3. P2 4. DDB 4. P1 5. AB 5. P3 3. DCBAAB→AB Statements Reasons 1. DCBAAB 1. GIVEN 2. DBCAAB 2. P1 3. BDCAAB 3. P1 4. BBB 4. P2 5. AB 5. P3 1. CBDCB→AB Statements Reasons 1. CBDCB 1. GIVEN 2. CBDBC 2. P1 3. CBC 3. P2 4. CCB 4. P1 5. AB 5. P3 2. ABCDCBA→AB Statements Reasons 1. ABCDCBA 1. GIVEN 2. ABA 2. P2 3. AAB 3. P1 4. AB 4. P3 4. CADCADCAD→AB Statements Reasons 1. CADCADCAD 1. GIVEN 2. ACDCADCAD 2. P1 3. ACDCADCDA 3. P1 4. ABA 4. P2 5. AAB 5. P1 6. AB 6. P3 5. ABBACCADD→AB Statements Reasons 1. ABBACCADD 1. GIVEN 2. ABBACCDAD 2. P1 3. ABBACCDDA 3. P1 4. ABA 4. P2 5. AAB 5. P1 6. AB 6. P3 If an animal is If an animal is
If an animal is If an animal is
not a mouse, not a rodent,
a mouse, then a rodent, then
then it is not a then it is not a
it is a rodent. it is a mouse.
rodent.
mouse.
If the sum of two
angles is 90°,
then the angles
are
complements of
each other.
If two angles are
complements of
each other, then
their sum is 90°.
If the sum of two If two angles are
angles is not
not
90°, then the
complements of
angles are not each other, then
complements of their sum is not
each other.
90°.
If two angles
are congruent,
then their
measures are
equal.
If two angles
have equal
measures,
then the
angles are
congruent.
If two angles
If two angles
are not
have unequal
congruent,
measures,
then their
then the
measures are angles are not
not equal.
congruent.
If m∠A = 28°
then ∠A is an
acute angle.
If ∠A is an
acute angle,
then
m∠A = 28°.
If ∠A is not an
If the measure
acute angle,
of angle A is
then the
not 28°, then
measure of
∠A is not an
angle A is not
acute angle.
28°.
If 2x – 3 = –7,
then x = –2.
If x = –2, then
2x – 3 = –7. If 2x – 3 ≠ –7,
then x ≠ –2.
If x ≠ –2, then
2x – 3 ≠ –7.
If a shape has
If a shape is a
four sides,
rectangle,
then the
then it has
shape is a
four sides.
rectangle.
If a shape is
not a
rectangle,
then the
shape does
not have four
sides.
If a shape
does not have
four sides,
then the
shape is not a
rectangle.
If you are a
teenager,
then you are
at least 13
years old.
If you are at
If you are not If you are less
least 13 years
a teenager,
than 13 years
old, then you then you must old, then you
are a
be less than
are not a
teenager.
13 years old.
teenager.
If she is
responsible
for the money,
then she is
guilty if it is
lost.
If she is not
If she is not
If she is guilty
responsible
guilty if the
if the money is
for the money, money is lost,
lost, then she
the she is not then she is not
is responsible
guilty if it is
responsible
for the money.
lost.
for the money.
If a number is
divisible by 2,
then it is
divisible by 4.
If a number is
divisible by 4,
then the
number is
divisible by 2.
If a number is
not divisible
by 2, then it is
not divisible
by 4.
If a number is
not divisible
by 4, then the
number is not
divisible by 2.
3B Symbolic Logic I. Conditional statements and Venn diagrams 1. Everyone living in Sarajevo is living in a war zone. Zlata is living in Sarajevo. Name__________________________________ 2. All hurricanes have winds above 75 mph. Hugo is a hurricane. Is the conclusion valid? 3. All plastic toys are unbreakable. This yellow truck 4. All plastic toys are unbreakable. This yellow truck is is unbreakable. Conclusion: The yellow truck is not plastic. Conclusion: The yellow truck is not plastic. unbreakable. The argument is invalid because the toy can be The argument is invalid because the toy can be unbreakable but not be plastic. unbreakable even though it is not plastic. Symbolic Logic implies: → and: Λ or: V not: ~ Translate into English if p = I read music, q = I play the piano, and r = I sing in the choir. 1. p→q If I read music, then I play the piano. 2. r→~q If I sing in the choir, then I do not play the piano. 3. ~(pΛr)→~q If I do not read music and sing in the choir, then I do not play the piano. 4. (pVq)→r If I read music or play the piano, then I sing in the choir. Create truth tables for the following: 5. ~p 6. pΛq 7. pVq 8. p→q 9. Under what conditions is ~pVq true? 10. Under what conditions is ~pΛ~q true? 11. Conditional: p→q 12. Converse: q→p
13. Inverse: ~p→~q 14. Contrapositive: ~q→~p 15. Mary says, “If I take my umbrella, I will not get wet.” a. Using p for “Mary takes her umbrella” and q for “Mary gets wet,” give a truth table for Mary’s statement. b. Suppose Mary forgets her umbrella but does not get wet. Was Mary’s statement a true statement? 16. The music professor says to Jerry, “If you don’t attend the concert, you get an F for the course.” a. Using p for “Jerry attends the concert” and q for “Jerry gets an F grade for the course,” give a truth table for the professor’s statement. b. Suppose Jerry attends the concert but gets an F in the course anyway. Was the professor’s statement a true statement? 17. Give a truth table for the expression p ∨ (~ p → q ) . ~P F F T T Q T F T F ~P→Q T T T F P T T F F P∨(~P→Q) T T T F 18. Use truth tables to determine if p ∧ ( p → q ) is equivalent to p ∧ q . P P→Q P∧P→Q P Q P∧Q T T T T T T T F F T F F F T F F T F F T F F F F They are equivalent because the outcomes are the same. 19. Use a truth table to prove that the following compound sentence is a tautology: ( p ∧ ( p → q ) ) → q
20. Draw the proper conclusion and draw the corresponding Venn diagram: If you live in Kobe, Japan, then you live in a quake-­‐stricken city. Toshima does not live in a quake-­‐stricken city. Conclusion: Toshima does not live in Kobe, Japan. 21. What is an equivalence relation? An equivalence relation ⇔ is a relation that satisfies the reflexive, symmetric, and transitive properties. For example, for any a, b, and c in a set S, the following properties hold true: Reflexive: a=a. Symmetric: If a=b, then b=a. Transitive: If a=b and b=c, then a=c. The truth of these statements implies that equality (=) is an equivalence relation. 22. Is “is taller than” an equivalence relation? Reflexive? No; Mike is not taller than himself. Symmetric? No; if Mike is taller than Jason, then Jason is not taller than Mike. Transitive? Yes; if Mike is taller than Jason and Jason is taller than William, then Mike is taller than William. 23. Is “is parallel to” an equivalence relation? Reflexive? No; line l is not parallel to itself. Symmetric? Yes; if line l is parallel to line m, then line m is parallel to line l. Transitive? Yes; if line l is parallel to line m, and line m is parallel to line n, then line l is parallel to line n. 24. Give an example of an equivalence relation. “is equal to”, “is congruent to”, “is similar to” 3/4 Mastering Angles Name______________________________________ 1. The following statement is true: If two angles are adjacent, then they share common vertex. What is the contrapositive of the given statement? If they do not share a common vertex, then the two angles are not adjacent. Is it true or false? True 2
2. Claim: If n -­‐1 is not divisible by 8, then n is even. Is it true? Or can you find a counterexample? True. 3. A conditional statement is given. Choose statements from the list and put them in the correct order to prove the conditional statement. You will not use all of the statements listed. Stuck path: A to G Correct path: D to B to C to F If Doug doesn’t wash the dog, then Miguel will go fishing Saturday. A. If Doug doesn’t wash the dog, then Doug’s sister Amanda will do it. B. If the dog makes the living room dirty, then Doug will have to stay home Saturday to clean. C. If Doug stays home Saturday to clean, then he won’t go to the baseball game with Miguel. D. If Doug doesn’t wash the dog, then the dog will make the living room dirty. E. If Miguel gets a new fishing pole, then he will go fishing Saturday. F. If Doug doesn’t go to the baseball game with Miguel, then Miguel will go fishing Saturday. G. If Doug’s sister Amanda washes the dog, then Doug will do the dishes for her on Monday. H. If Miguel gets paid for mowing lawns Friday, then he gets a new fishing pole. 4. If Alec washes the school’s windows, he will be paid $5.00 an hour. Alec washes the school’s windows for four hours and so gets paid $20.00. 5. If Maria leaves work at five o’clock, she will run into rush-­‐hour traffic. If Maria runs into rush-­‐hour traffic, she will arrive home in a bad mood. Therefore if Maria leaves work at five o’clock, she will arrive home in a bad mood. 6. If a politician decides to run for president, then he or she will make many visits to New Hampshire. Senator Dole has decided to run for president. Senator Dole will make many visits to New Hampshire. 7. The following statement is true: If an angle measures 48°, then it is acute. Which other statement must also be true? A. If an angle is acute, then it measures 48°. B. If an angle does not measure 48°, then it is not acute. C. If an angle is not acute, then it does not measure 48°. D. If an angle is acute, then it does not measure 48°. 8. Complements of the same angle are congruent. Conditional: If ∠1 and ∠2 are complementary and ∠2 and ∠3 are complementary, ∠1 and ∠3 are congruent. Given: Prove: Jumbled Statements 1. ∠1 ≅ ∠3 Jumbled Reasons 9. Definition of complementary angles 2. m∠1 = m∠3 10. Given 3. m∠1 + m∠2 = 90° 11. Reflexive Property of Equality 4. ∠2 and ∠3 are complementary angles 12. Substitution Property 5. m∠ 1 + m∠2 = m∠2 + m∠3 13. Definition of complementary angles 6. m∠2 + m∠3 = 90° 14. Subtraction Property of Equality 7. ∠1 and ∠2 are complementary angles 15. Given 8. m∠2 = m∠2 16. Definition of congruent angles 9. Conditional: If m∠GRA = m∠ERT, then m∠1 = m∠2. Given: Prove: Jumbled Statements Jumbled Reasons Statements 7. G
iven 2 1. m∠ERT = m∠3 + m∠2 8. Subtraction Property of Equality 1 2. m∠GRA = m∠ERT 3. m∠GRA = m∠1 + m∠3 9. Angle Addition Postulate 4. m∠1 + m∠3 = m∠3 + m∠2 10. Substitution Property 11. Angle Addition Postulate 5. m∠3 = m∠3 6. m∠1 = m∠2 12. Reflexive Property of Equality 3 4 5 6 Reasons 7 11 9 10 12 8 Statements 7 4 6 3 5 8 2 1 Reasons 10 15 9 13 12 11 14 16