Paper 2 Notes

Question 1

what is a statement?

Answer 1

must be true or false
* cannot be false, eg: non- statments : 5y+ 3= 14

Question 2

what is the truth value of a statement?

Answer 2

the true/ false outcome of a statement:
* eg: 2 is even truth value = true
* 2 is odd, truth value = false

Question 3

what does it mean to be logically equivalent?

Answer 3

the two statments have the same truth values in the same circumstances

Question 4

draw a truth table for
* not (A)
* (B) and (A)
* (B) or (A)

Question 5

Draw truth table for not A or B

Answer 5

• property of not is that it applies to what is immediately after : (not A ) or B

Question 6

Draw the truth table for if A then B

Answer 6

Same for:
B if A
A only if B

if you take out the trash, then you get a dollar.
If human, then mamal

Question 7

Draw the Venn Diagram for if A then B

Answer 7

Same for:
B if A
A only if B

(If you are in the shaded area it is true)

Question 8

Draw the truth table for A if and only if B (A iff B)

Answer 8

Same as
B iff A

Question 9

Draw the Venn Diagram for A iff B

Answer 9

Same as
B iff A

Question 10

Draw a combined truth table for :
If A then B
If B then A
A only if B
B only if A
A iff B
B iff A

Question 11

What is the converse of the following?

Answer 11

Just swap

Question 12

What is the contra positive of the following?

Answer 12

Swap the two and then negate

Question 13

What do these following statements become ?

Answer 13

Question 14

Draw the Diagram for A is sufficient for B

Answer 14

Sufficient means inside the venue diagram

Question 15

Draw the diagram for A is necessary for B

Answer 15

Necessary means the outer circle in the venn diagram

Question 16

Complete the statements and use notation

Answer 16

Question 17

What are the following statements equivalent to?

Answer 17

Question 18

What is the general structure for simple deductive proofs?

Answer 18

Question 19

What is the general structure for proof by contradiction?

Answer 19

Question 20

What is the general structure of proof by contra positive

Answer 20

Find the contrapositive of the statement and then prove this by direct proof (if … then)

Question 21

what are some problems that squaring introduces?

looking at proofs

Answer 21

squaring introduces additional solutions that must be checked

Question 22

what must always be done when square rooting?

Answer 22

putting a +- before the square root so that we are not elimating solutions

Question 23

what is wrong with diving expressions by some unknown?

Answer 23

what is the unknown = 0 then it will not be true

Question 24

what is wrong with squaring both sides of an inequality?

-5 < 3

Answer 24

will not always be true becuase what is one side of the inequality was negative

25< 9 ?

Question 25

if A > B then, sin A > sin B

Answer 25

cannot just take a function of both sides, this doesn’t always work

Question 26

Determine whether the following statements are true

Answer 26

Question 27

Determine whether the following statements are true

Answer 27

Question 28

What are other worded way to say these phrases (necessary and sufficient)?

Answer 28

Question 29

What about these following statements?

Answer 29

Question 30

Is being odd necessary or sufficient to be prime?

Answer 30

Not sufficient, coz eg:9
Not necessary, coz eg: 2

Question 31

What must you think / do when it comes up with the phrase “which of the following must be true /are necessarily true” ?

Answer 31

Try come up with counter examples that don’t use the condition yet still prove the statement correct

Question 32

What must you do when deciding whether something is sufficient? (If…. Then ..) format

Answer 32

Find a counter example for when the first part is true BUT then the second part is not true.

Question 33

How do you chose counter example for if then statements?

Answer 33

Must be true for the first if condition and then false for the then condition

Question 34

what are some tips to coming up with counterexamples?

Answer 34

1. go as trivial as possible
2. use edge cases ( negatives/ fractions/ decimals)