predicate logic

A farm consists of seven brown ducks, two

black ducks, six gray cows, ten black cows, five blue

buffalos, six yellow buffalos, and one black buffalo. Determine which of the following statements are true.

• There is an animal on the farm that is red

• Every animal on the farm is a duck or a

  mammal.

• Every animal on the farm is brown or

  gray or black.

• There is an animal on the farm that is

  neither a cow nor a duck.

• No animal on the farm is blue

Which of the following statements are true?

• Every integer is a real number.

• 0 is a positive real number.

• For every real number r, -r is a negative real

  number.

• Every real number is an integer.

• Some rational numbers are integers

Which of the following statements are false?

• $\forall x\in R,\ x\ge\frac{1}{x}$

• $\forall a\in Z,\ \frac{\left(a-1\right)}{a}$ is not an integer

• For all positive integer m and n, m*n >= m+n.

• For all real numbers x and y, $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$

Which one is the predicate statement of "All dinosaurs are extinct". Define P(x) to be "x is distinct", D is the set of dinosaur.

•  $\forall x\in D,\ P\left(x\right)$  

•  $\forall x\in D,\ \sim P\left(x\right)$  

•  $\forall x\in D,\ D\left(x\right)\rightarrow P\left(x\right)$  

•  $\exists x\in D,\ P\left(x\right)$  

Which one if the predicate statement of "All Java programs have at least 5 lines"

Let P(x) be "x is a Java programming", Q(x) be "x has at least 5 lines"

• $\forall x,\ P\left(x\right)\rightarrow Q\left(x\right)$

• $\forall x,\ P\left(x\right)\rightarrow\sim Q\left(x\right)$

• $\exists x,\ P\left(x\right)\rightarrow Q\left(x\right)$

• $\forall x,\ Q\left(x\right)\rightarrow P\left(x\right)$

Which one is the predicate statement of “The square of any rational number is rational”.

Let P(x) be "x is rational. Note that Q is the set of the rational set.

• $\forall x\in Q,\ P\left(x^2\right)$

• $\exists x\in Q,\ P\left(x^2\right)$

• $\forall x\in Q,\ P\left(x^{ }\right)$

• $\forall x\in Q,\ P\left(x\right)\ \rightarrow\ P\left(x^2\right)$

Consider the following statement:

 $\forall\ n\in\ Z,\$  if  $n^2$  is even then n is even

• All integers have even squares and are even

• Given any integer whose square is even, that integer is even

• For all integers, there are some whose square is even

• Any integer with an even square is even.

• All even integers have even squares.

Let x be a real number. Which of the following are true?

• x > 2 $\rightarrow$ 1 x > 1

• x > 2 $\rightarrow$ $x^2$ > 4

• $x^2$ > 4 $\rightarrow$ x > 2

• $x^2$ >4 $\leftrightarrow$ $\left|x\right|$ >2

What is the negation of the following statement (there may be more than one answer)?

"All football players are fast"

• There is a football player who is not fast.

• All football players are not fast

• There is a fast person who is not a football player.

• No football players are fast.

• Some football players are not fast.

What is a negation of the following statement?

"All cars have four doors"

• There are some cars that do not have four doors

• No cars have four doors

• Some cars have four doors

• All cars have two or three doors.