Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

Discrete Math

14. fve G functions 1 - 1 and cover. f(g-(x) = x ve f(x+ 2) = x + 3 since it is, how much is g (3)? A) 1 B) 2 C) 3 D) 4 E) 5

[D] Consider the function F: {1,5, 8} × {6, 10} → {0, 1, 2, 3, 4, 5} defined as follows: F: (a, b) → (4a + b)mod 6 (D.1) Fill in the ordered pairs in the diagram below to indicate the elements in the domain of F. Then draw in arrows to complete the diagram so that it correspondes to the function. ( • 0 • 1 • 2 • 3 • 4 • 5 (D.2) What is the range of F? (D.3) What is the co-domain of F? 3/5 (D.4) What is the preimage (inverse image) of 3? (D.5) What is the preimage (inverse image) of 0? (D.6) Is F one-to-one? If not provide a counterexample. (D.7) Is F onto? If not provide a counterexample.

Define a relation R from R to R as follows: For all (x, y) ER×R, (x, y) E R if, and only if, x = y² + 1. Prove that (17, -4) ER %3D

Show whether the relation below does or does not have the following properties and then state whether it is or is not an equivalence relation. • Reflexive • Irreflexive • Symmetric • Antisymmetric • Transitive

What kind of functions is the following funetion f(x) ? a) Even b) Odd c) Neither odd nor even d) None of them a b O d O e f O g O

A function F from A to B is a relation from A to B that satisfies two properties: 1. for every x€ A, there is a y E B such that E F 2. for all elements x E A and both y and z such that y E B and z E B: if (x,y) and (x,z) are in F, then

How many of the following functions are incremented for each x E R fi(x) = x2, f2(x) = -x², f3(x) = x, f¾(x) = -x, fs(x) = x³ = 33 - O A) 4 B) none C) 3 D) 2 O E) 1

1. Find these values. a) ⌊1.1⌋    b) ⌈1.1⌉       c) ⌊−0.1⌋         d) ⌈−0.1⌉       e) ⌈2.99⌉          f ) ⌈−2.99⌉ 2. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one. a) f(a) = b, f(b) = a, f(c) = c, f(d) = d b) f(a) = b, f(b) = b, f(c) = d, f(d) = c 3. Give an example of a function from N to N that is a) one-to-one but not onto. b) onto but not one-to-one. c) both onto and one-to-one (but different from the identity function). d) neither one-to-one nor onto