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2.1 Conditional Statements

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2.1 Conditional Statements

Conditional Statement – a logical statement with two parts.

P, the “if part”, hypothesis

Q, the “then part”, conclusion

If p, then q is symbolized as p -> q (p implies q).

Ex:

If you study hard, then you will get As.

If you live in Ridgewood, then you live in NJ.

If an angle is 28 degrees, then it is acute.

- All fruits have seeds.
- Two angles that add up to 180 degrees are supplementary.
- An even number is a number divisible by 2.
- 3x + 17 = 23, because x = 2.
- Only people who are 18 are allowed to vote.
- P=It is raining. Q =I must bring an umbrella.
- Today is Friday and tomorrow is Saturday.
Now go back and underline the hypothesis and circle the conclusions.

The negation of a statement is the

opposite of the original statement.

Symbol: ~pWords: not p

Examples: Find the negation of the statement.

- This suit is black.
- We are worthy.

Given Conditional Statement: p->q

Converse: Switch hypothesis and conclusion.

q->p

Inverse: Negate hypothesis and conclusion.

~p->~q

Contrapositive: Switch and negate the hypothesis and conclusion.

~q->~p

A counterexample is a case that fits the hypothesis but leads to a different conclusion.

You can prove a statement is false by finding a counterexample.

Practice finding counterexamples.

1) If a number is prime, then it is odd.

2) If you live in Ridgewood, you go to RHS.

3) If a food is round, then it is pizza.

4) If a 3D figure lacks vertices, it is a sphere.

Converse: If you are a musician, then you are a guitar player.T or F

Inverse: If you are not a guitar player, then you are not a musician. T or F

Contrapositive. If you are not a musician, then you are not a guitar player. T or F

Associate CONverse with SWITCHING the hypothesis and conclusion

No Strings Attached

- You SWITCH and NEGATE the hypothesis and conclusion.
- It is like taking the converse and inverse at the same time.

- A conditional statement and its contrapositive are either both true or false. They are equivalent statements.
- A converse statement and an inverse statement are also both true or false. They are also equivalent statements.
- But (for example), a conditional may be true and a converse may be false.

Everyone will have to share, so make sure you come up with something.

- Write the definition of perpendicular lines as a conditional statement. Is the converse true?
- Come up with your own conditional statement. Then find the converse, inverse and contrapositive.
- Write true or false next to each of the four statements you wrote in number 2.

When a conditional statement and its converse are true, you can write them as a single biconditional statement linking the hypothesis and conclusion with “if and only if”.

Words: p if and only if qSymbol: p <-> q

Write a biconditional statement for your conditional statement if both your conditional and converse were true. If not, write one for perpendicular lines.