Answer to Question #266483 in Discrete Mathematics for Mitra

Question #266483

Let ๐‘„(๐‘ฅ, ๐‘ฆ) denote "๐‘ฅ + ๐‘ฆ = ๐‘ฆโ€œ. What are the truth values of the quantifications โˆƒ๐‘ฆโˆ€๐‘ฅ๐‘„(๐‘ฅ, ๐‘ฆ) and โˆ€๐‘ฅโˆƒ๐‘ฆ๐‘„(๐‘ฅ, ๐‘ฆ) where the domain for all variables consists of all real numbers?



1
Expert's answer
2021-11-16T11:09:41-0500

1) โˆƒ๐‘ฆโˆ€๐‘ฅ๐‘„(๐‘ฅ, ๐‘ฆ)

Since for every y we can put x = 0, and then "0 + y = y\\implies y=y" , which means this statement is true

2) โˆ€๐‘ฅโˆƒ๐‘ฆ๐‘„(๐‘ฅ, ๐‘ฆ)

Since, for example, for x = 2 we have "2 + y = y\\implies 0=2" , which means this statement is false


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