Question

practice 2 - 1
show that each conditional is false by finding a counterexample.

1. if it is 12:00 noon, then the sun is shining.
2. if the car is full of gas, then the engine will start.
3. if a number is divisible by 3, then it is odd.

write the converse of each conditional.

1. if you drink milk, then you will be strong.
2. if a rectangle has four sides the same length, then it is a square.
3. if you do not sleep, you will be tired.

write the converse of each statement. if the converse is true, write true; if it is not true, provide a counterexample.

1. if (x - 4=22), then (x = 26).
2. if (|x|>0), then (x>0).
3. if (m^{2}) is positive, then (m) is positive.
4. if (y = 3), then (2y-1 = 5).
5. if point (a) is in the first - quadrant of a coordinate grid, then (x>0).
6. if two lines have equal slopes, then the lines are parallel.
7. if you are a twin, then you have a sibling.
8. draw a venn diagram to illustrate the statement in exercise 13.

practice 2 - 2
each conditional statement is true. consider each converse. if the converse is true, combine the statements and write them as a biconditional.

1. if two angles have the same measure, then they are congruent.
2. if (2x - 5=11), then (x = 8).
3. if (n = 17), then (|n|=17).
4. if a figure has eight sides, then it is an octagon.

write the two conditional statements that make up each biconditional.

1. a whole number is a multiple of 5 if and only if its last digit is either a 0 or a 5.
2. two lines are perpendicular if and only if they intersect to form four right angles.
3. you live in texas if and only if you live in the largest state in the

Explanation:

Step1: Analyze conditional 1

A counter - example for "If it is 12:00 noon, then the sun is shining" is a cloudy or rainy day at 12:00 noon.

Step2: Analyze conditional 2

A counter - example for "If the car is full of gas, then the engine will start" is a car with a dead battery even when full of gas.

Step3: Analyze conditional 3

A counter - example for "If a number is divisible by 3, then it is odd" is 6 (divisible by 3 but even).

Step4: Find converse of 4

The converse of "If you drink milk, then you will be strong" is "If you are strong, then you drink milk".

Step5: Find converse of 5

The converse of "If a rectangle has four sides the same length, then it is a square" is "If a figure is a square, then it is a rectangle with four sides the same length".

Step6: Find converse of 6

The converse of "If you do not sleep, you will be tired" is "If you are tired, then you did not sleep".

Step7: Analyze converse of 7

For "If \(x - 4=22\), then \(x = 26\)", the converse "If \(x = 26\), then \(x - 4=22\)" is true.

Step8: Analyze converse of 8

For "If \(|x|>0\), then \(x>0\)", the converse "If \(x > 0\), then \(|x|>0\)" is true. A counter - example for the original is \(x=- 1\) (\(| - 1| = 1>0\) but \(-1<0\)).

Step9: Analyze converse of 9

For "If \(m^{2}\) is positive, then \(m\) is positive", the converse "If \(m\) is positive, then \(m^{2}\) is positive" is true. A counter - example for the original is \(m=-1\) (\((-1)^{2}=1>0\)).

Step10: Analyze converse of 10

For "If \(y = 3\), then \(2y-1 = 5\)", the converse "If \(2y - 1=5\), then \(y = 3\)" is true. Solving \(2y-1 = 5\) gives \(2y=6\), \(y = 3\).

Step11: Analyze converse of 11

For "If point \(A\) is in the first quadrant of a coordinate grid, then \(x>0\)", the converse "If \(x>0\), then point \(A\) is in the first quadrant" is false. A counter - example is a point with \(x>0,y<0\) (fourth quadrant).

Step12: Analyze converse of 12

For "If two lines have equal slopes, then the lines are parallel", the converse "If two lines are parallel, then they have equal slopes" is true (excluding vertical lines).

Step13: Analyze converse of 13

For "If you are a twin, then you have a sibling", the converse "If you have a sibling, then you are a twin" is false.

Step14: Draw Venn - diagram for 13

Draw a large circle labeled "People with siblings". Inside it, draw a smaller circle labeled "Twins".

Step15: Analyze Practice 2 - 2, 1

For "If two angles have the same measure, then they are congruent", the converse "If two angles are congruent, then they have the same measure" is true. The biconditional is "Two angles are congruent if and only if they have the same measure".

Step16: Analyze Practice 2 - 2, 2

For "If \(2x-5 = 11\), then \(x = 8\)", the converse "If \(x = 8\), then \(2x-5 = 11\)" is true. The biconditional is "\(2x-5 = 11\) if and only if \(x = 8\)".

Step17: Analyze Practice 2 - 2, 3

For "If \(n = 17\), then \(|n|=17\)", the converse "If \(|n|=17\), then \(n = 17\)" is false (\(n=-17\) is a counter - example).

Step18: Analyze Practice 2 - 2, 4

For "If a figure has eight sides, then it is an octagon", the converse "If a figure is an octagon, then it has eight sides" is true. The biconditional is "A figure is an octagon if and only if it has eight sides".

Step19: Analyze Practice 2 - 2, 5

The two conditionals for "A whole number is a multiple of 5 if and only if its last digit is either 0 or 5" are: "If a whole number is a multiple of 5, then its last digit is either 0 or 5" and "If the last digit of a whole number is either 0 or 5, then it is a multiple of 5".

Step20: Analyze Practice 2 - 2, 6

The two conditionals for "Two lines are perpendicular if and only if they intersect to form four right angles" are: "If two lines are perpendicular, then they intersect to form four right angles" and "If two lines intersect to form four right angles, then they are perpendicular".

Step21: Analyze Practice 2 - 2, 7

The two conditionals for "You live in Texas if and only if you live in the largest state in the [assumed the US]" are false since Alaska is the largest state in the US. The conditionals are: "If you live in Texas, then you live in the largest state in the US" and "If you live in the largest state in the US, then you live in Texas".

Answer:

1. Counter - example: Cloudy/rainy 12:00 noon.
2. Counter - example: Dead battery with full gas.
3. Counter - example: 6.
4. Converse: If you are strong, then you drink milk.
5. Converse: If a figure is a square, then it is a rectangle with four sides the same length.
6. Converse: If you are tired, then you did not sleep.
7. Converse is true.
8. Converse is true, counter - example for original: \(x=-1\).
9. Converse is true, counter - example for original: \(m = - 1\).
10. Converse is true.
11. Converse is false, counter - example: Point with \(x>0,y<0\).
12. Converse is true (excluding vertical lines).
13. Converse is false.
14. Draw a large circle "People with siblings", inside it a smaller circle "Twins".
15. Biconditional: Two angles are congruent if and only if they have the same measure.
16. Biconditional: \(2x - 5=11\) if and only if \(x = 8\).
17. Converse is false (\(n=-17\)).
18. Biconditional: A figure is an octagon if and only if it has eight sides.
19. Conditional 1: If a whole number is a multiple of 5, then its last digit is either 0 or 5. Conditional 2: If the last digit of a whole number is either 0 or 5, then it is a multiple of 5.
20. Conditional 1: If two lines are perpendicular, then they intersect to form four right angles. Conditional 2: If two lines intersect to form four right angles, then they are perpendicular.
21. Conditional 1: If you live in Texas, then you live in the largest state in the US (false). Conditional 2: If you live in the largest state in the US, then you live in Texas (false).