Let ๐(๐ฅ, ๐ฆ) denote "๐ฅ + ๐ฆ = ๐ฆโ. What are the truth values of the quantifications โ๐ฆโ๐ฅ๐(๐ฅ, ๐ฆ) and โ๐ฅโ๐ฆ๐(๐ฅ, ๐ฆ) where the domain for all variables consists of all real numbers?
1) โ๐ฆโ๐ฅ๐(๐ฅ, ๐ฆ)
Since for every y we can put x = 0, and then "0 + y = y\\implies y=y" , which means this statement is true
2) โ๐ฅโ๐ฆ๐(๐ฅ, ๐ฆ)
Since, for example, for x = 2 we have "2 + y = y\\implies 0=2" , which means this statement is false
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