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Chapter 2: Problem 7

For all squares \(x\) there is a circle \(y\) such that \(x\) and \(y\) have different colors and \(y\) is above \(x\).

Short Answer

Expert verified
In summary, the given statement "For all squares \(x\) there is a circle \(y\) such that \(x\) and \(y\) have different colors and \(y\) is above \(x\)" is true. This is because there are no restrictions on the number of squares and circles, colors, or positions. Hence, it is always possible to find a circle with a different color positioned above a square.

Step by step solution

01

The statement given is: "For all squares x there is a circle y such that x and y have different colors and y is above x." We need to examine the conditions stated and determine if they hold true for all squares and circles. #Step 2: Dissect the conditions#

We can break down the statement's conditions into three main points: 1. There is always a circle for every square. 2. The circle and square have different colors. 3. The circle is positioned above the square. #Step 3: Check the first condition - There is always a circle for every square#
02

We need to determine if, for every square, there exists a circle. Since there are no limitations or restrictions provided on the number of squares and circles, we can assume that the possibility of having a circle for every square is indeed true. #Step 4: Check the second condition - The circle and square have different colors#

For this condition, we need to confirm if it's possible that all squares and circles have different colors. Since colors are not limited or specified in this problem, it is reasonable to assume that, whenever we have a square and circle pair, they could have different colors. #Step 5: Check the third condition - The circle is positioned above the square#
03

The last condition states that the circle must be above the square. In a two-dimensional plane, this is possible by having the circle's center at a higher vertical position than the square's top side. Since there are no position restrictions given, this condition can also be achieved in all cases. #Step 6: Conclude the proof#

Since all the conditions stated in the given statement hold true for all possible cases, we can conclude that for all squares x there is a circle y such that x and y have different colors, and y is above x.

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Most popular questions from this chapter

Write an informal negation for each of the following statements: a. All pots have lids. b. All birds can fly. c. Some pigs can fly, d. Some dogs have spots.

The following statement is true: " \(\forall\) nonzero numbers \(x, \exists\) a real number \(y\) such that \(x y=1 . .\) For each \(x\) given below, find a \(y\) to make the predicate " \(x y=1\) " true, a. \(x=2\) b. \(x=-1\) c. \(x=3 / 4\)

\(\forall n \in \mathbf{Z}\), if \(n\) is prime then \(n\) is odd or \(n=2\).

Some of the arguments in 7-18 are valid by universal modus ponens or universal modus tollens; others are invalid and exhibit the converse or the inverse error. State which are valid and which are invalid. Justify your answers. If a number is even, then twice that number is even. The number \(2 n\) is even, for a particular number \(n\). The particular number \(n\) is even.

Consider the following statement: \(\forall\) integers \(n\), if \(n^{2}\) is even then \(n\) is even. Which of the following are equivalent ways of expressing this statement? a. All integers have even squares and are even. b. Given any integer whose square is even, that integer is itself even. c. For all integers, there are some whose square is even. d. Any integer with an even square is even. e. If the square of an integer is even, then that integer is even. f. All even integers have even squares.
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