MAT 144 MIDTERM MAT 144 MIDTERM REVISION MATERIAL WITH QUESTIONS WITH VERIFIED ANSWERS (LATEST UPDATE 2024 ) GUARANTEED PASS!! Universal Statement says that a certain property is true for all elements in a set. (For example: All positive numbers are greater than zero.) Conditional Statement says that if one thing is t rue then some other thing also has to be true. (For example: If 378 is divisible by 18, then 378 is divisible by 6.) Existential Statement says that there is at least one thing for whic h the property is true. (For example: There is a prime number that is even.) Universal Conditional statement a statement that is both universal and conditional. (For example: for all animals a, if a is a dog, then a is a mammal.) ∀x, if P(x) then Q(x) MAT 144 MIDTERM Universal Existential Statement a statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something. (For example: Ever y real number has an additive inverse.) Existential Universal Statement a statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind. (For example: There is a positive integer that is less than or equal to every positive integer.) x ∈ S x is an element of the set S x ∈/ S x is not an element of the set S Set-roster notation when all elements are w ritten between two braces (ex: [1, 2, 3]) Axiom of extension says that a set is completely determined by what its elements are —not the order in which they might be listed or the fact that some elements might be listed more than once. R+ MAT 144 MIDTERM denotes the set of positive real numbers Z denotes the set of integers N denotes the set of natural numbers (nonnegative integers) Proper subset when every element of A is in B, but there is at least one element of A that is not in B Cartesian Prod uct of A and B "A cross B;" the set of all ordered pairs (a,b), where a is in A and b is in B. Relation R from A to B a subset of A × B; given an ordered pair (x, y) in A×B, x is related to y by R. The set A is called the domain of R and the set B is cal led its co -domain. Function F from set A to set B a relation with domain A and co -domain B that satisfies the following two properties: 1. For every element x in A, there is an element y in B such that (x, y) ∈ F. 2. For all elements x in A and y and z in B,
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