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# Exercises

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## Exercises

Use universal instantiation or universal modus ponents to complete the following arguments.

 If $a, b, c, d \in \mathbb{Z}$ and $b, d \ne 0$ then $\frac{a/b}{c/d} = \frac{ad}{bc}$$a=2, b=3, c=6, d=2$∴ ___________________________ All dogs can swim.John has a dog named Solar.∴ ___________________________ Differentiable functions are continuous.$f(x) = x^2 + 2x - 3$ is differentiable∴ ___________________________ If the lengths of the sides of a triangle satisfy the equation $a^2 + b^2 = c^2$ then the triangle is a right triangle.The sides of a triangle are 5, 12 and 13.∴ ___________________________

The following arguments are either valid or invalid. The valid arguments are examples of universal modue ponents or universal modus tollens; the invalid arguments are either converse errors or inverse errors. Determine which are valid or invalid and why.

 Every integer greater than 1 is divisible by a prime number.14 is an integer greater than 1.∴ 14 is divisible by a prime number. All rectangles have four sides.R is not a rectangle.∴ R doesn't have four sides. If $f:X\to Y$ and $g:Y\to Z$ are both one-to-one functions then $g \circ f$ is one-to-one.g is not one-to-one.∴ $g \circ f$ is not one-to-one. All birds can fly.John's pet can fly.∴ John's pet is a bird. All even numbers are divisible by 2.7 is not divisible by 2.∴ 7 is not even. If a function is differentiable then it's continuous.$f(x) = |x|$ is continuous.∴ $f(x)$ is differentiable. If G is a connected graph with integer n vertices and $n - 1$ edges then G is a tree.G is a graph with n vertices and is not a tree.∴ G doesn't have $n - 1$ vertices or is not connected. If x and y are real numbers and $xy=0$ then $x=0$ or $y=0$.$x=2$ and $xy=0$.∴ $y=0$.

Determine whether or not each of the following arguments is valid using a diagram.

 All polynomials are continuous.$f(x)$ isn't a polynomial.∴ $f(x)$ isn't continuous. All tall people are wealthyAll wealthy people are admired.Some basketball players are not admired.∴Some basketball players are not tall. All dogs can swim.No wolves can swim.∴ No wolves are dogs. All poets are young.Some young people are wise.John is a young poet.∴ John is not a wise. No restaurants serve fish.Fish is never expensive.∴ Some restaurants serve expensive food. All widgets are smooth.Some widgets are round.A round objects are green.∴ No round objects are smooth.

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