2.1-2.3 Quiz Review - Logic and Functions

A set of practice flashcards covering conjectures, function rules from ordered pairs, conditional statements, definitions, biconditionals, and basic logic reasoning from 2.1 to 2.3.

What is the method to show that a conjecture is false?

Find a counterexample.

Is the statement 'The sum of the squares of two consecutive squared natural numbers is even' true or false?

False. If the consecutive numbers are n^2 and (n+1)^2, their sum is n^2 + (n+1)^2 = 2n^2 + 2n + 1, which is odd.

Is the statement 'The sum of the squares of any two squared natural numbers is odd' true or false?

False. For example, 1^2 + 3^2 = 10 is even; the sum is not always odd.

For the ordered pairs (1,3), (2,4), (3,−5), (4,−6), what is a function rule relating x and y?

A piecewise rule: y = x + 2 for x ≤ 2; y = −x − 2 for x ≥ 3.

For the ordered pairs (1,4), (2,9), (3,16), (4,25), what is a function rule relating x and y?

y = (x + 1)^2.

Rewrite 'A car with leaking antifreeze has a problem' as an if-then statement.

If a car has leaking antifreeze, then it has a problem.

Rewrite 'Don't say anything at all when you don't have something nice to say' as an if-then statement.

If you don't have something nice to say, then you should not say anything.

Rewrite 'A vein is a blood vessel that carries blood toward the heart' as an if-then statement.

If something is a vein, then it carries blood toward the heart.

Rewrite the statement about a circle with radius r and circumference 2πr as a biconditional.

A circle has circumference 2πr if and only if its radius is r.

What are the two parts of a conditional statement?

Hypothesis and conclusion.

Define a conjecture.

An unproven statement that is based on observations.

Define a counterexample.

A specific case in which a given conjecture is false.

Define a polygon.

A closed plane figure formed by three or more sides with each side intersecting exactly two others at endpoints, and no two sides with a common endpoint are collinear.

Is the statement 'If a figure is an n-gon, then the figure is a polygon with n sides' a valid definition?

Yes; it describes the same concept in a definitional form.

Is the statement 'If a polygon is convex, has five sides, and is both equilateral and equiangular, then the polygon is a regular polygon' a valid definition?

Yes; such a polygon would be regular (a pentagon in this case).

Is the statement 'If a polygon is not convex, then it is a concave polygon' a valid definition?

Yes; for simple polygons, not convex implies concave.

Using Law of Detachment: What conclusion follows from 'If Dr. Klein is well-rested for a surgical procedure, then she operates with precision' and 'Dr. Klein got plenty of sleep to prepare for today's operation'?

Dr. Klein operates with precision.

Using Law of Detachment: If we don't make any stops, then we'll reach the stadium by 12:30 P.M. If we reach by 12:30, then we should be in time to see the kickoff. What conclusion follows?

If we don't make any stops, we should be in time to see the kickoff.

In a recreational basketball league, if a player receives two technical fouls in one game, then the player is ejected from the game. If a player is ejected from a game, then the player has to sit out the following game. What conclusion follows from two tech fouls?

The player is ejected from the game, and will sit out the following game.

If the accused suspect has a valid alibi, then the police will not hold him. The crime occurred at 9:32 A.M. The suspect's alibi places him with 22 other people from 7:00 A.M. to 4:00 P.M. on the same day. What conclusion follows?

The police will not hold him.

If firefighters enter a burning building, there is significant danger involved. A group of firefighters enters a burning warehouse to look for people. What follows?

There is significant danger involved.

Identify whether the reasoning is inductive or deductive: 'You diet for 3 weeks and lose 3 pounds. You conclude that you can lose 20 more pounds in the next 20 weeks.'

Inductive reasoning (generalizing from specific observations).