Explanation Given information: Consider that f : X → Y and g : Y → X are function such that g ∘ f = I X and f ∘ g = I Y then objective is to show that f and g are both one-to-one and onto and g = f − 1 . Concept used: A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain. A function is said to be onto function if each element in co-domain is mapped with atleast one element in domain. Proof: First show that f is one-to-one, for this consider f ( x 1 ) = f ( x 2 ) , where x 1 , x 2 ∈ X . Now, f ( x 1 ) = f ( x 2 ) This implies that, g ( f ( x 1 ) ) = g ( f ( x 2 ) ) Or, ( g ∘ f ) ( x 1 ) = ( g ∘ f ) ( x 2 ) Since g ∘ f = I X therefore, ( g ∘ f ) ( x 1 ) = ( g ∘ f ) ( x 2 ) implies that I X ( x 1 ) = I X ( x 2 ) . This implies that x 1 = x 2 This shows that f is one-to-one. For this consider y ∈ Y and objective is to show that there exists x ∈ X such that f ( x ) = y

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ISBN: 9781337694193

Author: EPP, Susanna S.

Publisher: Cengage Learning,

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Chapter 7.3, Problem 28ES

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| Without evaluating the Legendre symbols, prove the following. (i) 1(173)+2(2|73)+3(3|73) +...+72(72|73) = 0. (Hint: As r runs through the numbers 1,2,. (ii) 1²(1|71)+2²(2|71) +3²(3|71) +...+70² (70|71) = 71{1(1|71) + 2(2|71) ++70(70|71)}. 72, so does 73 – r.)

Chapter 7 Solutions

Discrete Mathematics With Applications

Ch. 7.1 - Given a function f from a set X to a set Y, f(x)...Ch. 7.1 - Given a function f from a set X to a set Y, if...Ch. 7.1 - Prob. 3TY Ch. 7.1 - Given a function f then a set X to a set Y, if...Ch. 7.1 - Prob. 5TY Ch. 7.1 - Prob. 6TY Ch. 7.1 - Prob. 7TY Ch. 7.1 - Prob. 8TY Ch. 7.1 - Prob. 9TY Ch. 7.1 - Prob. 1ES

Ch. 7.1 - Let X={1,3,5} and Y={a,b,c,d}. Define g:XY by the...Ch. 7.1 - Indicate whether the statement in parts (a)-(d)...Ch. 7.1 - a. Find all function from X={a,b}toY={u,v} . b....Ch. 7.1 - Let Iz be the identity function defined on the set...Ch. 7.1 - Find function defined on the sdet of nonnegative...Ch. 7.1 - Let A={1,2,3,4,5} , and define a function F:P(A)Z...Ch. 7.1 - Let Js={0,1,2,3,4} , and define a function F:JsJs...Ch. 7.1 - Define a function S:Z+Z+ as follows: For each...Ch. 7.1 - Prob. 10ES Ch. 7.1 - Define F:ZZZZ as follows: For every ordered pair...Ch. 7.1 - Let JS={0,1,2,3,4} ,and define G:JsJsJsJs as...Ch. 7.1 - Let Js={0,1,2,3,4} , and define functions f:JsJs...Ch. 7.1 - Define functions H and K from R to R by the...Ch. 7.1 - Prob. 15ES Ch. 7.1 - Let F and G be functions from the set of all real...Ch. 7.1 - Prob. 17ES Ch. 7.1 - Find exact values for each of the following...Ch. 7.1 - Prob. 19ES Ch. 7.1 - Prob. 20ES Ch. 7.1 - If b is any positive real number with b1 and x is...Ch. 7.1 - Prob. 22ES Ch. 7.1 - Prob. 23ES Ch. 7.1 - If b and y are positivereal numbers such that...Ch. 7.1 - Let A={2,3,5} and B={x,y}. Let p1 and p2 be the...Ch. 7.1 - Observe that mod and div can be defined as...Ch. 7.1 - Let S be the set of all strings of as and bs....Ch. 7.1 - Consider the coding and decoding functions E and D...Ch. 7.1 - Consider the Hamming distance function defined in...Ch. 7.1 - Draw arrow diagram for the Boolean functions...Ch. 7.1 - Fill in the following table to show the values of...Ch. 7.1 - Cosider the three-place Boolean function f defined...Ch. 7.1 - Student A tries to define a function g:QZ by the...Ch. 7.1 - Student C tries to define a function h:QQ by the...Ch. 7.1 - Let U={1,2,3,4} . Student A tries to define a...Ch. 7.1 - Prob. 36ES Ch. 7.1 - On certain computers the integer data type goed...Ch. 7.1 - Prob. 38ES Ch. 7.1 - Prob. 39ES Ch. 7.1 - Prob. 40ES Ch. 7.1 - Prob. 41ES Ch. 7.1 - In 41-49 let X and Y be sets, let A and B be any...Ch. 7.1 - Prob. 43ES Ch. 7.1 - Prob. 44ES Ch. 7.1 - Prob. 45ES Ch. 7.1 - Prob. 46ES Ch. 7.1 - Prob. 47ES Ch. 7.1 - Prob. 48ES Ch. 7.1 - Prob. 49ES Ch. 7.1 - Prob. 50ES Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.1 - Prob. 52ES Ch. 7.1 - Each of exercises 51-53 refers to the Euler phi...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - If F is a function from a set X to a set Y, then F...Ch. 7.2 - Prob. 3TY Ch. 7.2 - Prob. 4TY Ch. 7.2 - Prob. 5TY Ch. 7.2 - Prob. 6TY Ch. 7.2 - Prob. 7TY Ch. 7.2 - Given a function F:XY , to prove that F is not one...Ch. 7.2 - Prob. 9TY Ch. 7.2 - Prob. 10TY Ch. 7.2 - Prob. 11TY Ch. 7.2 - The definition of onr-to-one is stated in two...Ch. 7.2 - Fill in each blank with the word most or least. a....Ch. 7.2 - When asked to state the definition of one-to-one,...Ch. 7.2 - Let f:XY be a function. True or false? A...Ch. 7.2 - All but two of the following statements are...Ch. 7.2 - Let X={1,5,9} and Y={3,4,7} . a. Define f:XY by...Ch. 7.2 - Let X={a,b,c,d} and Y={e,f,g} . Define functions F...Ch. 7.2 - Let X={a,b,c} and Y={d,e,f,g} . Define functions H...Ch. 7.2 - Let X={1,2,3},Y={1,2,3,4} , and Z= {1,2} Define a...Ch. 7.2 - a. Define f:ZZ by the rule f(n)=2n, for every...Ch. 7.2 - Define F:ZZZZ as follows. For every ordered pair...Ch. 7.2 - a. Define F:ZZ by the rule F(n)=23n for each...Ch. 7.2 - a. Define H:RR by the rule H(x)=x2 , for each real...Ch. 7.2 - Explain the mistake in the following “proof.”...Ch. 7.2 - In each of 15-18 a function f is defined on a set...Ch. 7.2 - Prob. 16ES Ch. 7.2 - Prob. 17ES Ch. 7.2 - Prob. 18ES Ch. 7.2 - Referring to Example 7.2.3, assume that records...Ch. 7.2 - Define Floor: RZ by the formula Floor (x)=x , for...Ch. 7.2 - Prob. 21ES Ch. 7.2 - Let S be the set of all strings of 0’s and 1’s,...Ch. 7.2 - Define F:P({a,b,c})Z as follaws: For every A in...Ch. 7.2 - Les S be the set of all strings of a’s and b’s,...Ch. 7.2 - Let S be the et of all strings is a’s and b’s, and...Ch. 7.2 - Prob. 26ES Ch. 7.2 - Let D be the set of all set of all finite subsets...Ch. 7.2 - Prob. 28ES Ch. 7.2 - Define H:RRRR as follows: H(x,y)=(x+1,2y) for...Ch. 7.2 - Define J=QQR by the rule J(r,s)=r+2s for each...Ch. 7.2 - Prob. 31ES Ch. 7.2 - a. Is log827=log23? Why or why not? b. Is...Ch. 7.2 - Prob. 33ES Ch. 7.2 - The properties of logarithm established in 33-35...Ch. 7.2 - Prob. 35ES Ch. 7.2 - Prob. 36ES Ch. 7.2 - Prob. 37ES Ch. 7.2 - Prob. 38ES Ch. 7.2 - Prob. 39ES Ch. 7.2 - Suppose F:XY is one—to—one. a. Prove that for...Ch. 7.2 - Suppose F:XY is into. Prove that for every subset...Ch. 7.2 - Prob. 42ES Ch. 7.2 - Prob. 43ES Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - In 44-55 indicate which of the function in the...Ch. 7.2 - Prob. 46ES Ch. 7.2 - Prob. 47ES Ch. 7.2 - Prob. 48ES Ch. 7.2 - Prob. 49ES Ch. 7.2 - Prob. 50ES Ch. 7.2 - Prob. 51ES Ch. 7.2 - Prob. 52ES Ch. 7.2 - Prob. 53ES Ch. 7.2 - Prob. 54ES Ch. 7.2 - Prob. 55ES Ch. 7.2 - Prob. 56ES Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.2 - Write a computer algorithm to check whether a...Ch. 7.3 - If f is a function from X to Y’,g is a function...Ch. 7.3 - Prob. 2TY Ch. 7.3 - If f is a one-to=-one correspondence from X to Y....Ch. 7.3 - Prob. 4TY Ch. 7.3 - Prob. 5TY Ch. 7.3 - Prob. 1ES Ch. 7.3 - In each of 1 and 2, functions f and g are defined...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - In 3 and 4, functions F and G are defined by...Ch. 7.3 - Define f:RR by the rule f(x)=x for every real...Ch. 7.3 - Define F:ZZ and G:ZZ . By the rules F(a)=7a and...Ch. 7.3 - Define L:ZZ and M:ZZ by the rules L(a)=a2 and...Ch. 7.3 - Let S be the set of all strings in a’s and b’s and...Ch. 7.3 - Define F:RR and G:RZ by the following formulas:...Ch. 7.3 - Prob. 10ES Ch. 7.3 - Define F:RR and G:RR by the rules F(n)=3x and...Ch. 7.3 - The functions of each pair in 12—14 are inverse to...Ch. 7.3 - G:R+R+ and G1:RR+ are defined by G(x)=x2andG1(x)=x...Ch. 7.3 - H and H-1 are both defined from R={1} to R-{1} by...Ch. 7.3 - Explain how it follows from the definition of...Ch. 7.3 - Prove Theorem 7.3.1(b): If f is any function from...Ch. 7.3 - Prove Theorem 7.3.2(b): If f:XY is a one-to-one...Ch. 7.3 - Prob. 18ES Ch. 7.3 - If + f:XY and g:YZ are functions and gf is...Ch. 7.3 - If f:XY and g:YZ are function and gf is onto, must...Ch. 7.3 - Prob. 21ES Ch. 7.3 - If f:XY and g:YZ are functions and gf is onto,...Ch. 7.3 - Prob. 23ES Ch. 7.3 - Prob. 24ES Ch. 7.3 - Prob. 25ES Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1, and f1g1 , and...Ch. 7.3 - In 26 and 27 find (gf)1,g1,f1 , and f1g1 by the...Ch. 7.3 - Prob. 28ES Ch. 7.3 - Suppose f:XY and g:YZ are both one-to-one and...Ch. 7.3 - Prob. 30ES Ch. 7.4 - A set is finite if, and only if,________Ch. 7.4 - Prob. 2TY Ch. 7.4 - The reflexive property of cardinality says that...Ch. 7.4 - The symmetric property of cardinality says that...Ch. 7.4 - The transitive property of cardinality say that...Ch. 7.4 - Prob. 6TY Ch. 7.4 - Prob. 7TY Ch. 7.4 - Prob. 8TY Ch. 7.4 - Prob. 9TY Ch. 7.4 - Prob. 1ES Ch. 7.4 - Show that “there are as many squares as there are...Ch. 7.4 - Let 3Z={nZn=3k,forsomeintegerk} . Prove that Z and...Ch. 7.4 - Let O be the set of all odd integers. Prove that O...Ch. 7.4 - Let 25Z be the set of all integers that are...Ch. 7.4 - Prob. 6ES Ch. 7.4 - Prob. 7ES Ch. 7.4 - Use the result of exercise 3 to prove that 3Z is...Ch. 7.4 - Show that the set of all nonnegative integers is...Ch. 7.4 - In 10-14 s denotes the sets of real numbers...Ch. 7.4 - Prob. 11ES Ch. 7.4 - In 10-14 S denotes the set of real numbers...Ch. 7.4 - Prob. 13ES Ch. 7.4 - Prob. 14ES Ch. 7.4 - Show that the set of all bit string (string of 0’s...Ch. 7.4 - Prob. 16ES Ch. 7.4 - Prob. 17ES Ch. 7.4 - Must the average of two irrational numbers always...Ch. 7.4 - Prob. 19ES Ch. 7.4 - Give two examples of functions from Z to Z that...Ch. 7.4 - Give two examples of function from Z to Z that are...Ch. 7.4 - Define a function g:Z+Z+Z+ by the formula...Ch. 7.4 - âa. Explain how to use the following diagram to...Ch. 7.4 - Prob. 24ES Ch. 7.4 - Prob. 25ES Ch. 7.4 - Prove that any infinite set contain a countable...Ch. 7.4 - Prove that if A is any countably infinite set, B...Ch. 7.4 - Prove that a disjoint union of any finite set and...Ch. 7.4 - Prove that a union of any two countably infinite...Ch. 7.4 - Prob. 30ES Ch. 7.4 - Use the results of exercise 28 and 29 to prove...Ch. 7.4 - Prove that ZZ , the Cartesian product of the set...Ch. 7.4 - Prob. 33ES Ch. 7.4 - Let P(s) be the set of all subsets of set S, and...Ch. 7.4 - Prob. 35ES Ch. 7.4 - Prob. 36ES Ch. 7.4 - Prove that if A and B are any countably infinite...Ch. 7.4 - Prob. 38ES

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To prove or provide a counterexample for the given statement.

Solution Summary: The author explains that f:Xto Yandg:Y to X are function such that they are both one-to-one and onto.

To prove or provide a counterexample for the given statement.

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Chapter 7 Solutions